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Cornell University Library QA 155.B66 Introduction to higher algebra. Ililliiilliillllllllilllililllllilliillilllli 3 1924 002 750 473 Digitized by Microsoft® .« Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924002750473 INTEODUCTION TO HIGHER ALGEBRA THE MACMILLAN COMPANY NEW YORK • BOSTON • CHICAGO ■ DALLAS ATLANTA - SAN FRANCISCO MACMILLAN AND CO., Limited LONDON ■ BOMBAY - CALCUTTA - MADRAS MELBOURNE THE MACMILLAN COMPANY OF CANADA, Limited INTRODUCTION TO HIGHER ALGEBRA BY MAXIME BOCHER PBOFESSOK OF MATHEMATICS IN HARVARD UNIVEBSITY PREPARED FOR PUBLICATION WITH THE COOPERATION OF E. P. R. DUVAL INSTRUCTOR IN MATHEMATICS IN THE UNIVERSITY OF WISCONSIN PROPERTY OF LIBRARY NEW YORK STATE SCHOOL mWSTRIAL m LABOR RELATIONS CORNELL UNIVERSITY mm gotk THE MACMILLAN COMPANI Copyright, 1907, By the MACMILLAN COMPANY. All rights reserved — no part of this book may be reproduced in any form without permission in writing from the publisher, except by a reviewer who wishes to quote brief passages in connection with a review written for inclusion in magazine or newspaper. Set up and electrotyped. Published December, 190^. 45-15-1 » PRINTED IN THE UNITED STATES OF AMERICA • PREFACE An American student approaching the higher parts of mathe- matics usually finds himself unfamiliar with most of the main facts of algebra, to say nothing of their proofs. Thus he has only a rudimentary knowledge of systems of linear equations, and he knows next to nothing about the subject of quadratic forms. Students in. this condition, if they receive any algebraic instruction at all, are usually plunged into the detailed study of some special branch of algebra, such as the theory of equations or the theory of invariants, where their lack of real mastery of algebraic principles makes it almost inevitable that the work done should degenerate to the level of purely formal manipulations. It is the object of the present book to' introduce the student to higher algebraun such a way that he shall, on the one hand, learn what is meant by a proof in algebra and acquaint himself with the proofs of the most fundamental facts, and, on the other, become familiar with many important results of algebra which are new to him. The book being thus intended, not as a compendium, but really, as its title states, only as an introduction to higher algebra, the attempt has been made throughout to lay a sufficiently broad founda- tion to enable the reader to pursue his further studies intelligently, rather than to carry any single topic to logical completeness. No apology seems necessary for the omission of even such important subjects as Galois's Theory and a systematic treatment of invariants. A selection being necessary, those subjects have been chosen for treatment which have proved themselves of greatest importance in geametry and analysis, as well as in algebra, and the relations of the algebraic theories to geometry have been emphasized throughout. At the same time it must be borne in mind that the subject primarily treated is algebra, not analytic geometry, so that such geometric information as is given is necessarily of a fragmentary and some- what accidental character. No algebraic knowledge is presupposed beyond a familiarity with elementary algebra up to and including quadratic equations, and VI • PREFACE such a knowledge of determinants and the method of mathematical induction as may easily be acquired by a freshman in a week or two. Nevertheless, the book is not intended for wholly immature readers, but rather for students who have had two or three years' training in the elements of higher mathematics, particularly in analytic geometry and the calculus. In fact, a good elementary knowledge of analytic geometry is indispensable. The exercises at the ends of the sections form an essential part of the book, not merely in giving the reader an opportunity to think . for himself on the subjects treated, but also, in many cases, by sup- . plying him with at least the outlines of important additional theories. As illustrations of this we may mention Sylvester's Law of Nullity (page 80), orthogonal transformations (page 154 and page 173), and the theory of the invariants of the biquadratic binary form (page 260). On a first reading of Chapters I-VII, it may be found desirable to omit some or all of sections 10, 11, 18, 19, 20, 25, 27, 34, 35. The reader may then either take up the subject of quadratic forms (Chapte's VIII-XIII), or, if he prefer, he may pass directly to the more general questions treated in Chapters XIV-XIX. The chapters on Elementary Divisors (XX-XXII) form decid- edly the most advanced and special portion of the book. A person wishing to read them without reading the rest of the book should first acquaint himself with the contents of sections 19 (omitting Theorem 1), 21-25, 36, 42, 43. In a work of this kind, it has not seemed advisable to give many bibliographical references, nor would an acknowledgement at this point of the sources from which the material has been taken be feasible. The work of two mathematicians, however, Kronecker and Frobenius, has been of such decisive influence on the character of the book that it is fitting that their names receive special men- tion here. The author would also acknowledge his indebtedness to his colleague. Professor Osgood, for suggestions and criticisms relating to Chapters XIV-XVI. This book has grown out of courses of lectures which have been delivered by the author at Harvard University during the last ten years. His thanks are due to Mr. Duval, one of his former pupils, without whose assistance the book would probably never have been written. CONTENTS CHAPTER I POIYNOMIALS AND THEIR MoST FUNDAMENTAL PROPERTIES SUCTION p^g, 1. Polynomials in One Variable 1 ' 2. Polynomials in More than One Variable 4 3. Geometric Interpretations 8 4. Homogeneous Coordinates H 5. The Continuity of Polynomials 14 6. The Fundamental Theorem of Algebra 16 CHAPTER II A Few Properties of DBTBkMiNANTS 7. Some Definitions 20 8. Laplace's IJevelopraent . 24 9. The Multiplication Theorem 26 10. Bordered Determinants . 28 11. Adjoint Determinants and their Minors- 30 CHAPTER III The Theory of Linear Dependence 12. Definitions and Preliminary Theorems . 34 18. The Condition for Linear Dependence of Sets of Constants ... 36 14. The Linear Dependence of Polynomials 38 15. Geometric Illustrations 39 CHAPTER IV Linear Equations 16. Non -Homogeneous Linear Equations 43 17. Homogeneous Linear Equations 47 18. Fundamental Systems of Solutions of Homogeneous Linear Equations . 49 vii viii CONTENTS CHAPTER V Some Theorems Concerning the Rank of a Matrix BECTION pAex 19. General Matrices 54 20. Symmetrical Matrices 56 CHAPTER VI Linear Transformations and the Combination of Matrices 21. Matrices as Complex Quantities 60 22. The Multiplication of Matrices ......... 62 23. Linear Transformation . . . . 66 24. CoUineation 68 25. Further Development of the Algebra of Matrices 74 26. Sets, Systems, and Groups 80 27. Isomorphism 83 CHAPTER VII Invariants. First Principles and Illustrations 28. Absolute Invariants ; Geometric, Algebraic, and Arithmetical . . 8S 29. Equivalence 92 30. The Rank of a System of Points or a System of Linear Forms as an Invariant 94 31. Relative Invariants and Covariants 95 32. Some Theorems Concerning Linear Forms 100 33. Cross-Ratio and Harmonic Division 102 34. Plane-Coordinates and Contragredient Variables . . . • ' . 107 35. Line-Coordinates in Space 110 CHAPTER Vin Bilinear Forms 36. The Algebraic Theory 114 37. A Geometric Application 116 CHAPTER IX Geometric Introduction to the Study of Quadratic Forms 38. Quadric Surfaces and their Tangent Lines and Planes .... 118 39. Conjugate Points and Polar Planes 121 40. Classification of Quadric Surfaces by Means of their Rank . . . 123 41. Reduction of the Equation of a Quadric Surface to a Normal Form . 124 CONTENTS ■ ix CHAPTER X Quadratic Forms SBOTION PAOB 42. The General Quadratic Form and its Polar 127 43. The Matrix and the Discriminant of a Quadratic Form . . .' . 128 44. Vertices of Quadratic Forms 129 45. Reduction of a Quadratic Form to a Sum of Squares .... 131 46. A Normal Form, and the Equivalence of Quadratic Forms . . . 134 47. Reducibility 136 48. Integral Rational Invariants of a Quadratic Form 137 49. A Second Method of Reducing a Quadratic Form to a Sum of Squares . 139 CHAPTEE XI Real Quadratic Forms 50. The Law of Inertia 144 51. Classification of Real Quadratic Forms 147 52. Definite and Indefinite Forms 150 CHAPTEE XII The System of a Quadratic Form and Onb or More Linear Forms 53. Relations of Planes and Lines to a Quadric Surface 155 54. The Adjoint Quadratic Form and Other Invariants 159 55. The Rank of the Adjoint Form 161 CHAPTER XIII Pairs or QrADRAxic Forms 56. Pairs of Conies 163 57. Invariants of a Pair of Quadratic Forms. Their X-Equation . . . 165 58. Reduction to Normal Form when the X-Equation has no Multiple Roots 167 59. Reduction to Normal Form when i/» is Definite and Non-Singular . . 170 CHAPTEE XIV Some Properties op PoLYNOMiAiiS in General 60. Factors and Reducibility . ' 174 61. The Irreducibility of the General Determinant and of the Symmetrical Determinant 176 62. Corresponding Homogeneous and Non-Homogeneous Polynomials . . 178 X CONTENTS SECTION PAGE 63. Division of Polynomials 180 64. A Special Transformation of a Polynomial 184 CHAPTER XV Factors and Common Factors of Polynomials in One Variable AND OF Binary Forms 65. Fundamental Theorems on the Factoring of Polynomials in One Varia- ble and of Binary Forms 187 66. The Greatest Common Divisor of Positive Integers 188 67. The Greatest Common Divisor of Two Polynomials in One Variable . 191 68. The Resultant of Two Polynomials in One Variable .... 195 69. The Greatest Common Divisor in Determinant Form .... 197 70. Common Roots of Equations. Elimination 19S 71. The Cases a„=0 and J„ = 200 72- The Resultant of Two Binary Forms 201 CHAPTER XVI Factors of Polynomials in Two or More Variables 73. Factors Involving only One Variable of Polynomials in Two Variables . 203 74. The Algorithm of the Greatest Common Divisor for Polynomials in Two Variables 206 75. Factors of Polynomials in Two Variables 208 76. Factors of Polynomials in Three or More Variables 212 CHAPTER XVII General Theorems on Integral Rational Invariants 77. The Invariance of the Factors of Invariants 218 78. A More General Method of Approach to the Subject of Relative Invariants 220 79. The Isobaric Character of Invariants and Covariants .... 222 80. Geometric Properties and the Principle of Homogeneity .... 226 81. Homogeneous Invariants 230 82. Resultants and Discriminants of Binary Forms 236 CHAPTER XVIII Symmetric Polynomials 83. Fundamental Conceptions. 5 and S Functions 240 84. Elementary Symmetric Functions 242 85. The Weights and Degrees of Symmetric Polynomials .... 245 86. The Resultant and the Discriminant of Two Polynomials in One Variable 248 CONTENTS XI CHAPTER XIX Polynomials Symmetric in Pairs of Variables 6B0TI0N PAGE 87. Fundamental Conceptions. S and S Functions 252 88. Elementary Symmetric Functions of Pairs of Variables. . . . 253 89. Binary Symmetric Functions 255 90. Resultants and Discriminants of Binary Forms . . . . 257 CHAPTER XX Elementary Divisors and the Equivalence of X^Matrices 91. X-Matrices and their Elementary Transformations 262 92. Invariant Factors and Elementary Divisors ...... 269 93. The Practical Determination of Invariant Factors and Elementary Divisors 272 94. A Second Definition of the Equivalence of \-Matrices .... 274 95. Multiplication and Division of A.-Matrices 277 CHAPTER XXI The Equivalence and Classification of Pairs of Bilinear Forms and op Collineations 96. The Equivalence of Pairs of Matrices 279 97. The Equivalence of Pairs of Bilinear Forms 283 98. The Equivalence of Collineations 284 99. Classification of Pairs of Bilinear Forms 287 100. Classification of Collineations 292 CHAPTER XXII The Equivalence and Classification of Pairs of Quadratic Forms 101. Two Theorems in the Theory of Matrices 296 102. Symmetric Matrices 299 103. The Equivalence of Pairs of Quadratic Forms 302 104. Classification of Pairs of Quadratic Forms 305 105. Pairs of Quadratic Equations, and Pencils of Forms or Equations . . 307 106. Conclusion 31? Index • 31! mTRODUOTIO]^ TO HIGHER ALGEBRA CHAPTER I POLYNOMIALS AND THEIR MOST FUNDAMENTAL PROPERTIES 1. Polynomials in One Variable. By an integral rational func- tion of a;, or, as we shall say for brevity, a polynomial in a;, is meant a function of x determined by an expression of the form (1) c^a;"i + Cg*"* + ••• +"^2;''*, where the «'s aie integers positive or zero, while the c's are any con- stants, real or imaginary. We may without loss of generality assume that no two of the as are equal. This being the case, the expressions CiOf-i are called the terms of the polynomial, e^ is called the coefficient of this term, and a^ is called its degree. The highest degree of any term whose coefficient is not zero is called the degree of the polynomial. It should be noticed that the conceptions just defined — terms, coefficients, degree — apply not to the polynomial itself, but to the particular expression (1) which we use to determine the polynomial, land it would be quite conceivable that one and the same function of X might be given by either one of two wholly different expressions of the form (1). We shall presently see (cf. Theorem 5 below) that this cannot be the case except for the obvious fact that we may insert in or remove from (1) any terms we please with zero coefficients. By arranging the terms in (1) in the order of decreasing a's and supplying, if necessary, certain missing terms with zero coefficients, we may write the polynomial in the normal form (2) a^x^ + fflja;"-! + - + a^.^x + a„. B 1 2 INTRODUCTION TO HIGHER ALGEBRA It should, however, constantly be borne in mind that a polynomial in this form is not necessarily of the nth. degree ; but will be of the wth degree when and only when a^ ^ 0. Definition. Two polynomials, fiix) and f^ip^)-, o-'re said to he identically equal (/j =/2) i/" ^^^y ^'''^ equal for all values of x. A polynomial f{x) is said to vanish identically (/=0) if it vanishes for all values of x. We learn in elementary algebra how to add, subtract, and multi- ply * polynomials ; that is, when two polynomials f^ix) and f^ix) are given, to form new polynomials equal to the sum, difference, and product of these two. Theoeem 1. If the polynomial f{x) = a^x" + a^df-'^ + ...+a„ vanishes when x= a, there exists another polynomial ^1(2;) = a^x"-^ + aix''-^ + ■■■ + <_i, such that /(a;) = {x — a)(^j(a;). For since by hypothesis /( a) = 0, we have ^(.x)=f{x) -/(«) = %{^ - «") + «!(«"-' - «"-') + - + a„-i(a: - «). Now by the rule of elementary algebra for multiplying together two polynomials we have «* - «* =(a; - a) (a^-' + oa;*-" + • • • + a*-i). Hence /(a;) = (a;-«)[ao(a;»-» + aa;»-='H 1- «""') + aiCa;"-^ + aa;""' + — + «»-2)+... +a^_J. If we take as <^x(a;) the polynomial in brackets, our theorem is proved. Suppose now that yS is another value of x distinct from a for which "fix) is zero. Then /(/3) = (;8-«)<^i(;8)=0; * The question of division is somewhat more complicated and will be considered in § 63. i'OLYNOMIALS AND THEIR FUNDAMENTAL PROPERTIES 3 and since j3 — u^Q, ^^(/S) = 0. We can therefore apply tHe theo rem just proved to the polynomial <j>i{^), thus getting a new po ynomia ^^^^^ ^ ^^^^_^ _^ ^,,_^„_3 _j_ __, _|_ ^,,_^^ such that <^j(a;) = (x- 0)^^{x), and therefore f(x) = (a; — a) (a; — ^)4>^(z). Proceeding in this way, we get the following general result: Thboeem 2. If ui, a^i ••• % «*■« ^ distinct constants, and if f(x) = a^x" + aja;"-! + ... + a^ (w>i), aw«Z /(«!) =f{u^) = ••• =/(«*) = 0, then f{x) = {x- a{) (x - a^) ■■■{x- a^)4){x), where ^{x) = a^a:"-* + b-^x"-''-^ + — + 5„_i. Applying this theorem in particular to the case n = k,we see that if the polynomial f(x) vanishes for n distinct values aj, a^, ••• «„ of z, f{x) = a^{x - a^){x - ttj) ... (a; - «„). Accordingly, if a^^O, there can be no value of x other than «j, ••• «„ for which /(a;) = 0. We have thus proved Theorem 3. A polynomial of the nth degree in x cannot vanish for more than n distinct values of x. Since the only polynomials which have no degree are those all oi whose coefficients are zero, and since such polynomials obviously vanish identically, we get the fundamental result : Theorem 4. A necessary and sufficient condition that a polyno- mial in X vanish identically is that all its coefficients he zero. Since two polynomials in x are identically equal when and only when their difference vanishes identically, we have Theorem 5. A necessary and sufficient condition that two polyno mials in x he identically equal is that they have the same coefficients. This theorem shows, as was said above, that the terms, coefficients, and degree of a polynomial depend merely on the polynomial itself, not on the special way in which it is expressed. 4 INTRODUCTION TO HIGHER ALGEBRA 2. Polynomials in More than One Variable. A function of (2;, y^ is called a polynomial if it is given by an expression of the form c-^y^'^ + e^x^y^' + ••• + e^.x'^kyh, where the a's and /S's are integers positive or zero. More generally, a function of {x^, x^, ■•■ x„) is called a polynomial if it is determined by an expression of the form (1) e-^x-^x^-^ ■ ■ ■ x/^ + c^x-^x^'^ • • • a;/2 + • • • + c^x-^ux^i ■ ■ ■ a;/*, where the a's, /3's, •■■ v's are integers positive or zero. Here we may assume without loss of generality that in no two terms are the exponents of the various x's the same ; that is, that if then Vi =^ Vj. This assumption being made, CiX-^ix^*--- a;/* is called a term of the polynomial, e, its coefficient, a,- the degree of the term in x^, /3; in x^, etc., and a^ + yS, H h I'i the total degree, or simply the degree, of the terra. The highest degree in a;,- of any term in the polynomial ivhose coefficient is not zero is called the degree of the polynomial in Xi, and the highest total degree of any term whose coefficient is not zero is called the degree of the polynomial. Here, as in § 1, the conceptions just defined apply for the present not to the function itself but to the special method of representing it by an expression of the form (1). We shall see presently, how- ever, that this method is unique. Before going farther, we note explicitly that according to thi definition we have given, a polynomial all of ivhose coefficientu are zero has no degree. When we speak of a polynomial in n variables, we do not nee- essarily mean that all n variables are actually present. One or more of them may have the exponent zero in every term, and hence not appear at all. Thus a polynomial in one variable, or even a con- stant, may be regarded as a special case of a polynomial in any larger number of variables. A polynomial all of whose terms are of the same degree is said to be homogeneous. Such polynomials we will speak of as forms,* * There is diversity of usage here. Some writers, following Kronecker, apply the term /orm to all polynomials. On the other hand, homogeneous polynomials are often spoken of as guantics by English writer* POLYNOMIALS AND THEIR FUNDAMENTAL PROPERTIES 6 distinguishing between hinary, ternary, quaternary, and in general n-ary forms according to the number of variables involved, binarj forms involving two, ternary three, etc. Another method of classifying forms is according to their degree. We speak here of linear forms, quadratic forms, cubic forms, etc., according as the degree is 1, 2, 3, etc. We will,' how- ever, agree that a polynomial all of whose coefficients are zero may also be spoken of indifferently as a linear form, quadratic form, cubic form, etc., in spite of the fact that it has no degree. If all the coefficients of a polynomial are real, it is called a real polynomial even though, in the course of our work, we attribute imaginary values to the variables. It is frequently convenient to have a polynomial in more than one variable arranged according to the descending powers of some one of the variables. Thus a normal form in which we may write a polynomial in n variables is the <^'s being polynomials in the n — 1 variables {x^, ■•■ x„). We learn in elementary algebra how to add, subtract, and multiply polynomials, getting as the result new polynomials. Definition. Two polynomials in any number of variables are said to be identically equal if they are equal for all values of the vari- ables. A polynomial is said to vanish identically if it vanishes for all values of the variables. Theoreji 1. A necessary and sufficient condition that a polyno- mial in any number of variables vanish identically is that all its coeffi- cients be zero. That this is a sufficient condition is at once obvious. To prove that it is a necessary condition we use the method of mathematical induction. Since we know that the theorem is true in the case of one variable (Theorem 4, § 1), the theorem will be completely proved if we can show that if it is true for a certain number w — 1 of vari ables, it is true for n variables. Suppose, then, that vanishes identically. If we assign to (a^, ••• x„) any fixed values (x[, ■■•jc'„),f becomes a polynomial in Xi alone, which, by hypothesis. 6 INTRODUCTION TO HIGHER ALGEBRA vanishes for all values of Xy Hence its coefficients must, by Theorem 4, 8 1, all be zero: , , , r\ n /• n -i . ^\ That is, the polynomials <}>q, (^j, ••• <^m vanish for all values of the variables, since (x'^, ••■ x'„) was any set of values. Accordingly, by the assumption we have made that our theorem is true for polyno- mials in w — 1 variables, all the coefficients of all the polynomials <^o, <^i, ••■ ^OT are zero. These, however, are simply the coefficients of/. Thus our theorem is proved. Since two polynomials are identically equal when and only when their difference is identically zero, we infer now at once the further theorem : Theorem 2. A neceatary and sufficient condition that two poly- nomials be identically equal is that the coefficients of their corresponding terms be equal. We come next to Theorem 3. Iffi andfi are polynomials in any number ofvari-- ables of degrees rwj and m^ respectively, the product f^f^ will be of de- gree Wj + m^. This theorem is obviously true in the case of polynomials in one variable. If, then, assuming it tc be true for polynomials in ti — 1 variables we can prove it to be true ^or polynomials in n variables, the proof of our theorem by the method of mathematical induction will be complete. Let us look first at the special case in which both polynomials are homogeneous. Here every term we get by multiplying tftem together by the method of elementary algebra is of degree m^ + m^. Our theorem will therefore be proved if we can show that there is at least one term in the product whose coefficient is not zero. For this purpose, let us arrange the two polynomials /j and f^ according to descending powers of a;,, f{xi, ... x„)=<j)'i{x.i, ••• a;„)a^*« + ^{(a;2, - a;„)a^'i-»+ ..., Here we may assume that neither ^J nor ^H vanishes identically. Since /i and/j are homogeneous, <j>^ and <f>^' will also be homogeneous POLYNOMIALS AND THEIR FUNDAMENTAL PROPERTIES 7 of degrees Wj — k-^ and m^ — k^ respectively. In the product /j/j the terms of highest degree in x^ will be those in the product and since we assume our theorem to hold for polynomials in « — 1 variables, if>l, <(>(,' will be a polynomial of degree »»j + wij — Aj — /fe^. Any term in this product whose coefficient is not zero gives us when multiplied by a;j*i**a a term of the product /^/^ of degree mj^ + m^ whose coefficient is not zero. Thus our theorem is proved for the case of homogeneous polynomials. Let us now, in the general case, write /i and/j in the forms /iCa?!, ■■■x„)=(f>iX^, ■■■ a;„)+ (^4^_i(2;i, ...x„)+—, f 2(3:1, • • • a;„) = </)^(a;i, ■■■x„) + ^^-1(2^, ■■■x„)+—, where <^/ and <f>j' are homogeneous polynomials which are either of degrees i and j respectively, or which vanish identically. Since, by hypothesis, /j and f^ are of degrees nii and m2 respectively, ^^, and <j>ll^ will not vanish identically, but will be of degrees wij and m^. The terms of highest degree in the product fif^ will therefore be the terms of the product (f>l„^ 0^, and this being a product of homo- geneous polynomials comes under the case just treated and is there- fore of degree nii + m^. The same is therefore true of the product /1/2, and our theorem is proved. By a successive application of this theorem we infer Corollary. If k polynomials are of degrees mj, m^, ••• w^ re- spectively, their product is of degree m^ + m^-i h WJ^. We mention further, on account of their great importance, the two rather obvious results : Theorem 4. If the product of two or more polynomials is identi- cally zero, at least one of the factors must he identically zero. For if none of them were identically zero, they would all have definite degrees, and therefore their product would, by Theorem 3, have a definite degree, and would therefore not vanish identically. It is from this theorem that we draw our justification for cancel- ling out from an identity a factor which we know to be not identi- cally zero. 8 INTRODUCTION TO HIGHER ALGEBRA Theorem %. Iff(x^, ■ ■ ■ x„) is a polynomial which is not identically zero, and if cf>{x^, •■■ x„) vanishes at all points where f does not vanish, then <f) vanishes identically. This follows from Theorem 4 when we notice that fcf} =0. EXERCISES 1. If / and c^ are polynomials in any number of variables, what can be inferred from the identity /^ = <^^ concerning the relation between the polynomials / and <ji1 2. If /j and f^ are polynomials in (Xj, ••• x„) which are of degrees m^ and m, respectively in xi, prove that their product is of degree m^ + m2 in x^ 3. Geometric Interpretations. In dealing with functions of a single real variable, the different values which the variable may- take on may be represented geometrically by the points of a line ; it being understood that when we speak of a point x we mean the point which is situated on the line at a distance of x units (to the right or left according as x is positive or negative) from a certain fixed origin 0, on the line. Similarly, in the case of functions of two real variables, the sets of values of the variables may be pictured geometrically by the points of a plane, and in the case of three real variables, by the points of space ; the set of values represented by a point being, in each case, the rectangular coordinates of that point. When we come to functions of four or more variables, however, this geometric representation is impossible. The complex variable x= ^ + r]i depends on the two independent real variables | and 77 in such a way that to every pair of real values (}, T)) there corresponds one and only one value of x. The different values which a single complex variable may take on may, therefore, be represented by the points of a plane in which (^, 7?) are used as cartesian coordinates. In dealing with functions of more than one complex variable, however, this geometric representation is impos- sible, since even two complex variables x = ^-fr]i, y = ^ + r] i are equivalent to four real variables (f, r], ^j, t^j). By the neighborhood of a point x=a we mean that part of the line between the points x=a — u and x= a + ,t{a being an arbitrary positive constant, large or small), or what is the same thing, all points whose coordinates x satisfy the inequality \x — a\<u.* *We use the symbol \Z \ to denote the absolute value of Z, i.e. the numericai value of Z if Z is real, the modulus of Z if Z is imaginary. POLYNOMIALS AND THEIR FUNDAMENTAL PROPERTIES 9 Similarly, by the neighborhood of a point (a, b) in a plane, we shall mean all points whose coordinates (x, y) satisfy the inequalities \x-a\<a, |«/-5|</3, where a and /8 are positive constants. This neighborhood thus con- sists of the interior of a rectangle of which (a, 6) is the center and whose sides are parallel to the coordinate axes. By the neighborhood of a point (a, 6, c) in space we mean all points whose coordinates {x, y, z) satisfy the inequalities \x—a\<u, |y — 6|</3, |2— c|<7. In all these cases it will be noticed that the neighborhood may be large or small according to the choice of the constants If we are dealing with a single complex variable x = ^ + rji, we understand by the neighborhood of a point a all points in the plane of complex quantities whose complex coordinate x satisfies the in- equality \x— a\<K, a being as before a real positive constant. Since \x — a\ is equal to the distance between x and a, the neighborhood of a now consists of the interior of a circle of radius a described about a as center. It is found convenient to extend the geometric terminology we have here introduced to the case of any number of real or complex variables. Thus if we are dealing with n independent variables {x-^, x^, ■■■ x„), We /speak of any particular set of values of these variables as a point in space of n dimensions. Here we have to distinguish between real points, that is sets of values of the x's which are all real, and imaginary points in which this is not the case. In using these terms we do not propose even to raise the question whether in any geometric sense there is such a thing as space of more than three dimensions. We merely use these terms in a wholly conventional algebraic sense because on the one hand they have the advantage of conciseness over the ordinary algebraic terms, and on the other hand, by calling up in our minds the geometric pictures of three dimensions or less, this terminology is often suggestive of new relations which might otherwise not present themselves to us so readily. 10 INTRODUCTION TO HIGHER ALGEBRA By the neighborhood of the point (a^, a^, ••• a„) we understand all points which satisfy the inequalities \Xi-ai\<«i, {x^-a^lKa^-- |a;„-a„|<a„, where Kj, a^, ■•• a„ are real positive constants. If, in particular, (aj, aSj, ••• «„) is a real point, we may speak of the real neighborhood of this point, meaning thereby all real points {x^, x^,---x„) which satisfy the above inequalities. As an illustration of the use to which the conception of the neighborhood of a point can be put in algebra, we will prove the following important theorem : Theorem 1. A necessary and sufficient condition that a poly- nomial /(ajj, ••• x„) vanish identically is that it vanish throughout the neighborhood of a point (a^, ••• a„). That this is a necessary condition is obvious. To prove that it is sufficient we begin with the case n = 1. Suppose then that /(a;) vanishes throughout a certain neighbor- hood of the point x=a. \if{x) did not vanish identically, it would be of some definite degree, say k, and therefore could not vanish at more than h points (cf. Theorem 3, § 1). This, however, is not the case, since it vanishes at an infinite number of points, namely all points in the neighborhood of x = a. Thus our theorem is proved in the case w = 1. Turning now to the case « = 2, let fix, y) = ,l>,{y)x' + 4>i{y)x'-' +■■'+ Ut/) be a polynomial which vanishes throughout a certain neighborhood of the point (a, J), say when \x-a\<a, \y-b\<ff. Let y^ be any constant satisfying the inequality l2/o-5|<i8. Then f(x, y^) is a polynominal in x alone which vanishes whenever la;— a| < «. Hence, by the case w = 1 of our theorem, f(x, y^) = 0. That is. POLYNOMIALS AND THEIR FUNDAMENTAL PROPERTIES 11 Thus all these polynomials tj) vanish at every point y^ in the neigh- borhood of y = 6, and therefore, by the case « = 1 of our theorem, they are all identically zero. From this it follows that for every value of X, f(z, y) vanishes for all values of y, that is /= 0, and our theorem is proved. We leave to the reader the obvious extension of this method of proof to the case of n variables by the use of mathematical induction. From the theorem just proved we can infer at once the following: Theorem 2. A necessary and sufficient condition that two polyno- mials in the variables (a;^, •••a;„) be identically equal is that they be equal throughout the neighborhood of a point {a^, ■ ■ • a„). EZintCISES 1. Theorem 3, § 1 may be stated as follows : If / is a polynomial in one variable which is known not to be of degree higher than n, then if / vanishes at n + 1 distinct points, it vanishes identically. Establish the following generalization of this theorem : If /is a polynomial in (x, y) which is known not to be of higher degree than n in X, and not of higher degree than m in y, then, if / vanishes at the (n + 1) (m + 1) distinct points : / i = 1, 2, ... n + l\ (^>' 36') V ;■= l' 2 ••■ m + 1/' it vanishes identically. ■' ' '"' 2. Generalize the theorem of Exercise 1 to polynomials in any number of variables. 3. Prove Theorem 4, § 2 by means of Theorem 1 of the present section ; and from this result deduce Theorem 3, § 2. 4. Do Theorems 1 and 2 of this section hold if we consider only real polyno- mials and the real neighborhoods of real points ? 4. Homogeneous Coordinates. Though only two quantities are necessary in order to locate the position of a point in a plane, it is frequently more convenient to use three, the precise values of the quantities being of no consequence, but only their ratios. We will represent these three quantities by x, y, t, and define their ratios by the equations ^ — yr H— V t t where X and Y are the cartesian coordinates of a point in a plane. Thus (2, 3, 5) will represent the point whose abscissa is | and whose ordinate is f . Any set of three numbers which are proportional to 12 INTRODUCTION TO HIGHER ALGEBRA (2, 3, 5) will represent the same point. So that, while to every set of three numbers (with certain exceptions to be noted below) there corresponds one and only one point, to each point there cot- respond an infinite number of different sets of three numbers, all of which, however, are proportional. When ^ = our definition is meaningless ; but if we consider the points (2, 3, 1), (2, 3, 0.1), (2, 3, 0.01), (2, 3, 0.001), •••, which are, in cartesian coordinates, the points (2, 3), (20, 30), (200, 300), (2000, 3000), •••, we see that they all lie on the straight line through the origin whose slope is |. Thus as t approaches zero, x and «/ remaining fixed but not both zero, the point {x, y, t) moves away along a straight line through the origin whose slope is y/x. Hence it is natural to speak of (a;, y, 0) as the point at infinity on the line whose slope is y/x. If t approaches zero through negative values, the point will move off along the same line, but in the opposite direction. We will not distinguish between these two cases, but will speak of only one point at infinity on any particular line. It can be easily verified that if a point moves to infinity along any line parallel to the one just considered, its homogeneous, coordinates may be made to approach the same values (a;, y, 0) as those just obtained. It is therefore natural to speak of the point at infinity in a cer- tain direction rather than on a definite line. Finally we will agree that two points at infinity whose coordinates are propoitional shall be regarded as coinciding, since these coordinates may be regarded as the limits of the coordinates of one and the same point which moves further and further off.* \i x = y = t = Q, we will not say that we have a point at all, since the coordinates of any point whatever may be taken as small as we please, and so (0, 0, 0) might be regarded as the limits of the coor- dinates of any fixed or variable point. * It should be noticed that in speaking of points at infinity we are, considering the matter from a purely logical point of view, doing exactly the same thing that we did in § 3 in speaking of imaginary points, or points in space of n dimensions ; that is, we are speaking of a set of quantities as a "point" which are not the coordinates of any point. The only difference between the two cases is that the coordinates of our "point at infinity " are the limits of the coordinates of a true point. Thus, in particular, it is a pure convention, though a desirable and convenient one, when we say that two points at infinity shall be regarded as coincident when and only when their coordinates are proportional. We might, if we chose, regard all points at infinity as coincident. There is no logical compulsion in the matter. POLYNOMIALS AND THEIR FUNDAMENTAL PROPERTIES 13 The equation AX^ + BXY+ CY^+ DX+EY+ F= becomes, in homogeneous coordinates, or Aay^ + Bxy + Oy^ + Bxt + Eyt + M^ = 0, a homogeneous equation of the second degree ; and it is evident that if the coordinates JT, Y in any algebraic equation be replaced by the coordinates a;, «/, i, the resulting equation will be homogeneous, and of the same degree as the original equation. It is to this fact that the system owes its name, as well as one of its chief advantages. The equation Ax + By+Qt=^ represents, in general, a line, but if J. = ^= 0, C^ 0, it has no true geometric locus. It is, in this case, satisfied by the coordinates of all points at infinity, and by the coordinates of no other point. We shall therefore speak of it as the equation of the line at infinity. The reader may easily verify, by using the equation of a line in terms of its inter- cepts, that if a straight line move further and further away, its homo- geneous equation will approach more and more nearly the form ^=0. In space of three dimensions we will represent the point whose cartesian coordinates are X, Y, Z by the four homogeneous coordi- nates X, y, z, t, whose ratios are defined by the equations f = X, ^=F, '- = Z. t t t We will speak of (x, y, e, 0) as " the point at infinity " on a line whose direction cosines are X y g Va? + y^ + 2^' V«2 + ^2 + «2' Va;2 -H «/2 + 22 (0, 0, 0, 0) will be excluded, and t = will be spoken of as the equa tion of the plane at infinity. Extending the same terminology to the general case, we shall sometimes find it convenient to speak of {xy, x^, ■•■ x„) not as a point in space of n dimensions, but as a point represented by its homo- geneous coordinates in space of w — 1 dimensions. Two points 14 INTRODUCTION TO HIGHER ALGEBRA whose coordinates are proportional will be spoken of as identical, a point whose last coordinate is zero will be spoken ^i ,1^ a, point at infinity, and the case a^= ••• =x„=0 will not be spoken of as a point at all. This terminology will be adopted only in connection with homogeneous polynomials, and even then it must be clearly understood that we are perfectly free to adopt whichever terminol- ogy we find most convenient. Thus, foF instance, if /(xy x^, x^) is a homogeneous polynomial of the second degree, the equation /= may be regarded either as determining a conic in a plane (x-^, a;^, x^ being homogeneous coordinates) or a quadric cone in space (a;j, x^, r^ being ordinary cartesian coordinates). Homogeneous coordinates may also be used in space of one dimension. We should then determine the points on a line by two coordinates x, t whose ratio x/t is the non-homogeneous coordinate X, i.e. the distance of the point from the origin. It is this repre- sentation that is commonly made use of in connection with the theory of binary forms. 5. The Continuity of Polynomials. Definition. A function f{xi, ■■■ x„) is said to be continuous at the point (cj, • • • c„) if, no matter how small a positive quantity e be chosen, a neighborhood of the point (cj, •■■ c„) can be found so small that the dif- ference between the value of the function at any p nnt of this neighbor hood and its value at the point (cj, ••• c„) is in absolute value less than e. That is, / is continuous at (cj, •■■ c„) if, having chosen a positive quantity e, it is possible to determine a positive S such that \f{xi, ■■■x„)-f(ci,...e„)\<e for all values of (2^ • • • x„) which satisfy the inequalities, \xi-c,\<B, \Xi-c,\ < S, ... \x„-eJ<S. Theorem 1. If two functions are continuous at a point, their sum is continuous at this point. Let/i and/2 be two functions continuous at the point (cj, ■■■ c„] and let ki and A, be their respective values at this point. Then, no matter how small the positive quantity e may be chosen, we may take Si and Sj so small that 1/1 - *i I < J e when 1 2;< - c, | < Sj, |/2-Aj|<Je when I a;< - Ci I < Sj. POLYNOMIALS AND THEIR FUNDAMENTAL PROPERTIES 16 Accordingly l/i-*il + l/8-*2l<e when I a;< - (?< I < S, where B is the smaller of the two quantities Sj and Sj ; and, since |A| + |J5|^|A + 5|, we have l/i - *i +A - *2l = l(/i +/2) - (*i + ^2)1 < e when \xt - Ci\ < S. Hence /i +/,, is continuous at the point (c^, ••• c„). CoROLLABY. If a finite number of functions are continuous at a point, their sum is continuous at this point. Theorem 2. ^ two functions are continuous at a point, their product is continuous at this point. Let/j and/2 be the two functions, and k^ and k^ their values at the point (cj, ••• <?„) where they are assumed to be continuous. We have to prove that however small e may be, S can be chosen so small that (1) 1/1/2 -*i*2l<e when |a;i-c,|<S. Let ij be a positive constant, which we shall ultimately restrict to a certain degree of smallness, and let us choose two positive constants «! and Sj, such that |/i - ^i I < »? when | a;, - c, | < Sj, 1/2 -h\<V when |a;j - c^I < Sg. Now take S as the smaller of the two quantities S, and S^. Then, when \xi — c,|<S, 1/1/2 - Ai*2l = \Mfi - *i) + h(A - h)\ ^ I/2I 1/1- *il + 1*111/2-^1 ^J|/2l + l*lN- Accordingly since, when ja^j — c,| < S, 1/2! = 1*2 + (/2 - *2)l ^ 1*2! + 1/2 - *2| < 1*2! + V, we may write (2) |/i/2-*l*2|<{|*ll + l*2ll'; + '?'- If ki and k^ are not both zero, let us take 17 small enough to satisfy the two inequalities .^ 16 INTRODUCTION TO HIGHER ALGEBRA If ^1 = ^2 = 0, we will restrict 7} merely by the inequality i7<Ve. In either case, inequality (2) then reduces to the form (1) and oui theorem is proved. Corollary. If a finite number of functions are continuous at a point, their product is continuous at this point. Referring now to our definition of continuity, we see that any constant ma<, be regarded as a continuous function of (a;j, • • • x„) for all values of these variables, and that the same is true of any one of these variables themselves. Hence by the last corollary any function Ox the form Ca^> ■•• a^'", where the k's are integers positive or zero, is continuous at every point. If we now refer to the corollary to Theorem 1, we arrive at the theorem: Theorem 3. Ani/ polynomial is a continuous function for all values of the variables. Finally, we give a simple application of this theorem. Theorem 4. If f(xi, ••■ a;„) is a polynomial and f{c.^, ■■■ e„)g!=0, it is possible to take a neighborhood of the point (cj, • • ■ c„) so small that f does not vanish at any point in this neighborhood. Let k=f(ci, ••• e„). Then, on account of the continuity of/ at (<?!> ••• c„), a positive quantity S can be chosen so small that through- out the neighborhood |a;, — c,! < S, the inequality \f-^<m is satisfied. In this neighborhood / cannot vanish; for at any point where it vanished we should have \f-k\=m<m^ which is impossible since by hypothesis k^O. 6. The Fundamental Theorem of Algebra. Up to this point no use has been made of what is often known as the fundamental theorem of algebra, namely the proposition that every algebraic equation has a root. This fact we may state in more precise form as follows : POLYNOMIALS AND THEIR FUNDAMENTAL PKOPERTIES IT Theorem 1. If f{x) is a polynomial of the nth degree where « ^ 1, there exists at least one value of xfor which f{x) = 0. This theorem, fundamental though it is, is not necessary for most of the developments in this book. Moreover, the methods of proving the theorem are essentially not algebraic, or only in part algebraic. Accordingly, we will give no proof of the theorem here, but merely refer the reader who desires a formal proof to any of the text-books on the theory of functions of a complex variable. We shall, however, when we find it convenient to do so, assume the truth of this theorem. In this section we will deduce a few of its more immediate consequences. Theorem 2. If f(x) is a polynomial of the nth degree, fix) = af^x" + a^a;"-! + - +a„_iX + a„ (a„ # 0), there exists one and only one set of constants, a^, a^, ••• «„, such that f{x)=a^{x - a^{x -a^-{x- «J. This theorem is seen at once to be true for polynomials of the first degree. Let us then use the method of mathematical induction and assume the proposition true for all polynomials of degree less than n. If we can infer that the theorem is true for polynomials of the nth. degree, it follows that being true for those of the first degree it is true for those of the second, hence for those of the third, etc. By Theorem 1 we see that there is at least one value of x for which f{x) = 0. Call such a value Uy By Theorem 1, ^ 1 we may write f(x) = (x-ai)^{x), where 4>(x) = a^x"-'^ + b^stf-^ H + b„_i. Since <f)(x) is a polynomial of degree n — 1, and since we are assum- ing our theorem to be true for all such polynomials, there exist n — 1 constants a^, •■• «„ such that <ji(x) = a^{x - «2) - (a; - «„). Hence , f(x) = aQ{x— ai){x — a2) — i^ — «^' Thus half of our theorem is proved. 18 INTRODUCTION TO HIGHER ALGEBRA Suppose now there were two such sets of constants, Wj, •■• <*» aQ<J ySj, ••• /S„. We should then have (1) /(*) = «o(* - «i) - (a; - «„) s ao(x - ^{) — {x - /3,). Let a; = efi in this identity. This gives «o(«i-^i)(«i-A)-(«i-/3„) = 0. Accordingly, since a^^O, aj must be equal to one of the quan- tities ySj, /Sj, ••• /3„. Let us suppose the /3's to have been taken in such an order that a^ = /Sj. Now in the identity (1) cancel out the factor aQ{x — Wj) (see Theorem 4, § 2). This gives {x - a^) — (x- «„) = {x- /Jj) —(x- /3,). Accordingly, since we have assumed the theorem we are proving to be true for polynomials of degree n— 1, the constants ySj, ••• /8„ are the same, except perhaps for the order, as the constants Oj, ••• «„, and our theorem is proved. Definition. 77ie constants «j, ••• «„ determined in the last theo- rem are called the roots of the polynomial f{x), or of the equation f(x) = 0. If k of these roots are equal to one another, hut different from all the other roots, this root is called a kfold root. It is at once seen by reference to Theorem 1, § 1 that these roots are the only points at which /(a;) vanishes. Theorem 3. Iff{xy, ••• a;„) is a polynomial which is not identi- cally equal to a constant, there are an infinite number of points (a;,, ••• x„) at which f=^0, and also an infinite number at which f=0, provided w > 1. The truth of the first part of this theorem is at once obvious, for, since /is not identically zero, a point can be found at which it is not zero, and then a neighborhood of this point can be taken so small that /does not vanish in this neighborhood (Theorem 4, § 5). This neighborhood, of course, consists of an infinite number of points. To prove that / vanishes at an infinite number of points, let us select one of the variables which enters into / to at least the first degree. Without loss of generality we may suppose this variable to be Xy We may then write f{x^. - x„)^F^{x^, ... a;„)a;f + J'i(aj„ ••. a;,)a;J-' + ... + F^ix^, ... «;„), POLYNOMIALS AND THEIR FUNDAMENTAL PROPERTIES 19 where k>l and Fq is not identically zero. Let (<?j, ••• c„) be any point at which F^ is not zero. Then/(a;i, c^, ••• o„) is a polynomial of the Ath degree in x^ alone. Accordingly, by Theorem 1, there is at least one value of x^ for which it vanishes. If Cj is such a value, /(Cj, Cj, ••• c„) = 0. Moreover, by the part of our theorem already proved, there are an infinite number of points where F^ =!=■ 0, that is an infinite number of choices possible for the quantities Cj, ••■ <?„. Thus our theorem is completely proved. Finally, we will state, without proof, for future reference, a theorem which says, in brief, that the roots of an algebraic equation are continuous functions of the coefficients : Theorem 4. If a is a root of the polynomial then no matter how small a neighborhood |a;— a|<e of the point a we may consider, it is possible to take in space of n+1 dimensions a neigh- lorhood of the point {aQ,ai, ■■■ a„) so small that, if (bQ,b^, — 6J is any point in this neighborhood, the polynomial b^af + 6ia;»-i + - + b„_iZ + b„ has at least one root fi in the neighborhood \x—a\<€ of the point a. For a proof of this theorem we refer to Weber's Algebra, Vol. 1, § 44. • The theorem remains true if we merely assume that the polynomial is of at least the first degree. That is, some of the first coefficients oo, oi, ••• may he zero. CHAPTER n A FEW PROPERTIES OF DETERMINANTS 7. Some Definitions. We assume that the reader is familiar with the determinant notation, and will merely recall to him that by a determinant of the nib. order *21 *nl "ni nn *2» we understand a certain homogeneous polynomial of the wth degree in the n^ elements a,j-. By the side of these determinants it is often desirable to consider the system of the n^ elements arranged in the order in which they stand in the determinant, but not combined into a polynomial. Such a square array of n^ elements we speak of as a matrix. In fact, we will lay down the following somewhat more general definition of this term : Definition 1. A system of mn quantities arranged in a rectangu- lar array of m rows and n columns is called a matrix. If m = n, we say that we have a square matrix of order n. It is customary to place double bars on each side of this array, thus: "n *21 *12 *22 *ln \a, 'ml *m2 20 A FEW PROPERTIES OF DETERMINANTS Sometimes parentheses are used, thus : 21 Even when a matrix is square, it must be .carefully noticed that it is not a determinant. In fact, a matrix is not a quantity at all,* but a system of quantities. This difference between a square ma- trix and a determinant is clearly brought out if we consider the effect of interchanging columns and rows. This interchange has no effect on a determinant, but gives us a wholly new matrix. In fact, we will lay down the definition: Definition 2. Two square matrices »u M *lre '11 •*1» ^»1 of which either is obtained from the other hy interchanging rows and jolumns are called conjugate f to each other. Although, as we have pointed out, square matrices and deter- minants are wholly different things, every determinant determines a square matrix, the matrix of the determinant, and conversely every square matrix determi*""* a determinant, the determinant of the matrix. Every matrix contains other matrices obtained from it by strik- ing out certain rows or columns or both. In particular it contains certain square matrices ; and the determinants of these square matrices we will call the determinants of the matrix. If the matrix contains m rows and n columns, it will contain determinants of all orders from 1 (the elements themselves) to the smaller of the two integers and m and n inclusive. J In many important problems all * Cf., however, § 21. t Sometimes also transposed. t if TO = re, there is only one of these determinants of highest order, and it was this which, ws called above the determinant of the square matrix. 22 INTRODUCTION TO HIGHER ALGEBRA of these determinants above a certain order are zero, and it is often of great importance to specify the order of the highest non-vanish- ing determinant of a given matrix. For this purpose we lay down the following definition : Definition 3. A matrix is said to be of rank r if it contains at least one r-rowed determinant which is not zero, while all determinants of order higher than r which the matrix may contain are zero. A matrix is said to he of rank if all its elements are zero. For brevity, we shall speak also of the rank of a determinant, meaning thereby the rank of the matrix of the determinant. We turn now to certain definitions concerning the minors of determinants ; that is, the determinants obtained from the given determinant by striking out certain rows and columns. It is a familiar fact that to every element of a determinant corresponds a certain first minor; namely, the one obtained by striking out the row and column of the determinant in which the given element lies. Now the elements of a determinant of the nth order may be regarded as its (n — l)th minors. Accordingly we have here a method of pairing off each one- rowed minor of a given determinant with one of its (n — l)-rowed minors. Similarly, if ilf is a two-rowed minor of a determinant of the «th order i), we may pair it off against the (» — 2)-rowed minor N obtained by striking out from 2) the two rows and columns which are represented in M. The two minors M and N we will speak of as complementary. Thus, in the determinant the two minors *21 *81 are complementary. hi *31 »61 *U *jj *32 *43 '♦ea "IS *M 43 "it •14 *24 *34 *44 *64 ^K "U «ia «: '42 '68 14 *44 *S4 »M *46 *6S A FEW P&OPERTIES OP DETERMINANTS 23 In the same way we pair off with every three-rowed minor an (» — 3)-rowe(i minor; etc. In general we lay down D'BPINITION 4. If D 18 a determinant of the nth order and M one of its k-rowed minors, then the (n — k)-rowed minor N obtained hy striking out from D all the rows and columns represented in M is called the complement of M. Conversely, Mis clearly the complement of N. Let us go back now for a moment to the case of the one-rowed minors ; that is to the elements themselves. Let a^,- be the element of the determinant D which stands in the ith row and the yth column. Let Dij represent the corresponding first minor. It will ^e recalled that we frequently have occasion to consider not this minor D^ but the cof actor Ay of a^- defined by the equation J^ = (— ly+^'i)^. Similarly, it is often convenient to consider not the complement of a given minor but its algebraic complement, which in the case just mentioned reduces to the cofactor, and which, in general, we define as follows : Definition 5. If M is the m-rowed minor of D in which the rows ^n ••• ^m ^f^^ *^* columns li, ■■• l^ are represented, then the algebraic comr plement of M is defined by the equation alg. compl. of il!f=(-l)*'+"+*"'+''+"+'"'[compl. of ilf]. The following special case is important : Definition 6. By a principal minor of a determinant D is under- stood a minor obtained by striking out from D the same rows as columns. Since in this case, using the notation of Definition 5, we have it follows that the algebraic complement of any principal minor is equa to its plain complement. We have so far assumed tacitly that the orders of the minors wo were dealing with were less than the order n of the determinant itself. By the n-rowed minor of a determinant D of the rath order we of course understand this determinant itself. The complement of this minor has, however, by our previous definition no meaning. We will define the complement in this case to be 1, and, by Definition 5, this will also be the algebraic complement. 24 INTRODUCTION TO HIGHER ALGEBRA EXERCISE Prove that, if M and N- are complementary minors, either M and N are ths algebraic complements of each other, or — iV is the algebraic complement of M and — ikf is the algebraic complement of N. 8. Laplace's Development. Just as the elements of any row or column and their corresponding cofactors may be used to develop a determinant in terms of determinants of lower orders, so the A-rowed minors formed from any k rows or columns may be used, along with their algebraic complements, to obtain a more general development of the determinant, due to Laplace, and which includes as a special case the one just referred to. In order to establish this develop- ment, we begin with the following preliminary theorem : ^ Theorem 1. If the rows and columns of a determinant D he shifted in such a way as to bring a certain minor M into the upper left- hand corner without changing the order of the rows and columns either of M or of its complement N, then this shifting will change the sign of D or leave it unchanged according as — N or N is the algebraic com- plement of M.y To prove this let us, as usual, number the rows and columns of J), beginning at the upper left-hand corner, and let the numbers of the rows and columns represented in M, arranged in order of increas- ing magnitude, be A^, •••A^, and l-^, •••?„ respectively. _ In order to effect the rearrangement mentioned in the theorem, we may first shift the row numbered k^ upward into the first position, thus carry- ing it over k^ — 1 other rows and therefore changing the sign of the determinant k^^ — l times. Then shift the row numbered k^ into the second position. This carries it over Aj — 2 rows and hence changes the sign Ajg — 2 times. Proceed in this way until the row numbered k^ has been shifted into the mth position. Then shift the columns in a similar manner. The final result is to multiply -Z) by. Comparing this with Definition 5, § 7, the truth. of our theorem is obvious. , Lemma. If M is a minor of a determinant A the product of M by its algebraic complement is identical, when expanded, with some of the terms of the expansion of D. A FEW PROPERTIES OF DETERMINANTS 25 Let 2) = "m *nl and call the order of M, m, and its complement iV.^ We will first prove our lemma in the special case in which M stands in the upper leftrhand corner of D, so that iV, which in this case is the algebraic complement, is in the lower right-hand corner. \ What we have to show here is that the product of any tdrm oi Moj a, term, of iV is a term of D, and that this term does not come in twice to the product MM y Any term of M may be written where the integers Zj, l^-i-'-tm are merely some arrangement of the integers 1, 2, •••m, and fj, is the number of inversions of order in this arrangementA Similarly, any term of N may be written where Im+v •" 4 is merely some arrangement of the integers m-r-1, •■■ n, and v is the number of inversions of order in this arrangement^ The product of these two terms is a term of B, for the factors a are chosen in succession from the first, second, ••• nth rows of J), and no two are from the same col- umn,' and fi+ vis clearly precisely the number of inversions of order in' the arrangement Zj, l^, ■•■ l„,a.s compared to the natural arrange- ment, 1, 2, ••■ n, of these integers.'^ Having thus proved our lemma in the special case in which M lies in the upper left-hand corner of D, we now pass to the general case. ^^ Here we may, by shifting rows and columns, bring M into the upper left-hand corner and JV into the lower right-hand corner. This has, by Theorem 1, the effect of leaving each term in the expansion of D unchanged, or of reversing the sign of all of them according as iVor — iVis the algebraic complement of M.\ Accord- ingly, since the product il!fi\r gives, as we have just seen, terms in the expansion of this rearranged determinant, the product of M by its algebraic complement gives terms in the expansion of D itself. as was to be proved.x 26 INTRODUCTION TO HIGHER ALGEBRA Laplace's Development, which may be stated in the form of the following rule, now follows at once : Theorem 2. Pick out any m rows (or columns) from a determi- nant J), and form all the m-rowed determinants from this matrix.^ The sum of the products of each of these minors by its algebraic complement is the value of D. Since, by our lemma, each of these products when developed con- sists of terms of 2), it remains merely to show that every term of D occurs in one and only one" of these products.- ^ This is obviously the case ; for every term of D contains one element from each of the m rows of D from which our theorem directs us to pick out «i-rowed determinants, and,/since these elements all lie in different columns, they lie in one and only one of these wi-rowed determinants, say l£^ Since the other elements in this term of D obviously all lie in the complement N of M, this term will be found In the product MN and in none of the other products mentioned in our theorem._^ v \ EXERCISES 1. From a square matrix of order n and rank r, s rows (or columns) are selected. Prove that the rank of t}ie matrix thus obtained cannot be less than r -ir s — n. 2. Generalize the theorem of Exercise 1. 9. The Multiplication Theorem. Laplace's Development enables us to write out at once the product of any two determinants as a single determinant whose order is the sum of the orders of the two given determinants «fll — «1« ^1 — hm «nl ••• «»» Kl •'• 5mm ■ Pm, ^ml ■ whatever the values of the ^'s may be. For, expanding the large determinant in terms of the n-rowed minors of the first n rows, all the terms of the expansion are zero except the one written in the first member of the equation. A FEW PROPERTIES OF DETERMINANTS 27 «1 «a «8 A ^2 ^8 • 7i 7a 78 «1 «2 «3 /3i /32 /Sa 7i 73 7s -1 «1 «2 «s -1 h J. h -1 Ci Co e. From this formula we will now deduce a far more important one for expressing the product of two determinants of the same order as a determinant of that order. For this purpose let us choose the p'a in the last formula as follows : Pij = when i ^J, p^ = - 1, and let us consider for simplicity the product of two determinants of the third order. We have Let us now reduce this six-rowed determinant by multiplying its first column by a^ and adding it to the fourth column; then multiply the first column by a^ and add it to the fifth; then multiply the first column by a^ and add it to the sixth. In this way we bring zeros into the last three places in the fourth row. Next multiply the second column successively by 6j, Jj, 63 and add it to the fourth, fifth, and sixth columns respectively. Finally multiply the third column successively by Cj, c^, Cg and add it to the fourth, fifth, and sixth columns. The determinant thus takes the form a^ «j ag aiai+a^h^ + ttgCi aiaj+a^Jj + agCj «i«8+«2^3+"8<'3 7i 7a 7s 7i«i + 7a*i + 78<'i 7ia2 + 72*2 + 78^2 7i«8 + 72*8 + 73«8 -10 0-100 0-1,0 and this reduces at once to the three-rowed determinant «!«! -t- «,Ji + BjCi ai«2 -f- «2&2 -f- ttjCa tti^g + «2*S + H^z /3iai-|-y8jJi + /33C, ^^a^^ ^ih^ + ^^e^ ^1 flg + ^2 *3 + ^8''8 7i«i + 7a^i + 78''i 7i«a + 72^a + 78<'3 7i«8 + 7a^j. + 73«8 28 INTRODUCTION TO HIGHER ALGEBRA We have thus expressed the product of two determinants of the third order as a single determinant of the third order. The methbd we have used is readily seen to be entirely general, and we thus get the following rule for multiplying together two determinants of the nth order : Theorem. The product of two determinants of the nth ordm- may he expressed as a determinant of the nth order in which the element which lies in the ith row and jth column is obtained hy multiplying each element of the ith row of the first factor by the corresponding element of the jth column of the second factor and adding the results. It should be noted that changing rows into columns in either or both of the given determinants, while not affecting the value of the product, will alter its form materially. For example, 2 3 1 7 20 41 4 5 6 9 34 73 2 3 1 6 23 39 4 6 7 9 39 69 2 4 1 T 26 501 3 5 • 6 9 33 66 2 4 1 6 80 48 3 6 7 9 38 63 = 66, = 66, = 66; and similarly the product of any two determinants of the same ordei may be written in four different forms. 10- Bordered Determinants. If to a determinant of the wth order we add one or more rows and the same number of columns of n quantities each and fill in the vacant corner with zeros, the result- ing determinant is called a bordered determinant. Thus starting from the two-rowed determinant /3 A FEW PROPERTIES OF DETERMINANTS 29 we may form the bordered determinants a /8 Ui 7 S Ma 5 ^1 ^2 a /3 "l 7 8 "2 ^1 ^2 < < a ^ "l «i «(' 7 S "2 ni K "i "2 < < .i' t-i' If in tlie second of these examples we use Laplace's Development to expand the bordered determinant according to the two-rowed determinants of the last two rows, we see that its value is «1 < ^1 ^zl M3 < V'l v'.l a quantity into which the elements «, /3, 7, S of the original deter- minant do not enter. Similarly expanding the third of the above bordered determinants according to the three-rowed determinants of its last three rows, we see that its value is zero. The reasoning we have here used is of general application and leads to the following results : Theorem 1. If a determinant of the nth order is bordered with n rows and n columns, the resulting . determinant has a value which depends only on the bordering quantities. Theorem 2. If a determinant of the nth order is bordered with more than n rows and columns, the resulting determinant always has the value zero. The cases of interest are therefore those in which the deter- minant is bordered with less than n rows and columns. Concerning these we will establish the following fact : Theorem 3. If a determinant of the nth order be bordered by p rows and p columns (p < n) of independent variables, the resulting determinant is a polynomial of degree 2p in the bordering quantities, whose coefficients are the pth minors of the original determinant; and conversely, every pth minor of the original determinant is the coefficient of at least one term of this polynomial. 30 INTRODUCTION TO HIGHER ALGEBRA Let us consider the special case where m = 4 and p = 2. D = '11 »21 *31 *12 *22 *8a 42 *13 *23 *88 '48 *U *24 *84 *44 «4 M «18 «M «1 .Mi «1 ^2 «28 «24 "2 Mi + • • to 6 terms "l «2 «88 «48 «84 «44 Ms «4 < Developing this determinant, by Laplace's method (§ 8), in terms of the two-rowed determinants of the last two rows, we have D= If now we expahd each of these four-rowed determinants, by Laplace's method, in terms of the two-rowed determinants of their last two columns, and then arrange the result as a polynomial in the tt's and d's, the truth of the theorem is apparent. We leave it to the reader to fill in the details of the proof here sketched. 11. Adjoint Determinants and their Minors. Definition. If, in the determinant 2) = *nl A^ is the eof actor of the element a^, then the determinant *i« D'= is called the adjoint of J^ -^u ■■■ -^Xn ■^nl A FEW PROPERTIES OF DETERMINANTS 31 By corresponding minors of 2) and D', or indeed of any two determinants of the same order, we shall naturally understand minors obtained by striking out the same rows and columns from 2) as from B'. These definitions being premised, the fundamental theorem here is the following : , Theorem. If D' is the adjoint of any determinant B, and M and M' are corresponding m-rowed minors of D and D' respectively, then M' is equal to the product of I)^~^ hy the algebraic complement of M.\^ We will prove this theorem first for the special case in which the minors iHf and M lie at the upper left-hand corners of D and D' respectively. We may then write M' = Ax • •' -^im ... . •• A» A^, . A ... •• A^ .. 1 . . .. 1 . .. • • .. • . . 1 > Let us now interchange the columns and rows of 2), D = *n n» *nl \ and then form the product M'B by the theorem of § 9. This gives Z> — — D — M'D = »!» a. J) l.m+l "2.«»+l a, ■m,m+l «*m+l,m+l *i»+l,» = 2)" *m+l,m+l "* *n,m+l 'm+l,» > 3 \ 32 INTRODUCTION TO HIGHER ALGEBRA Let us here regard a^p ■■■ «„„ as n^ independent variable^! Then the equation just written becomes an identity, from which 17 since it is not identically zero, may be cancelled out, and we get (1) M'-B'"-^ ^m+l,m+l '" ^n,m+l Since the determinant which is written out in (1) is precisely the algebraic complement of M, our theorem is proved in the special case we have been considering. ' It should be noticed that this proof holds even in the case m = n; cf. Corollary 2 below^ . Turning now to the case in which the minors iHf and M' do not lie at the upper left-hand corners of D and D', let us denote by a the sum of the numbers which specify the location of the rows and columns in M ov M', the numbering running, as usual, from the up- per left-hand corner. \ Then by Definition 5, § 7, (2) alg. compl. of M={- If [compl. of il!f].\ Let us now, by shifting rows and columns, bring the determinant M into the upper left-hand corner of B. Calling the determinant D, as thus rearranged, D^ we have (cf. Theorem 1, § 8) (3) D,= (-iyi).y The cof actors in D^ are equal to (— 1)M^-, since the interchange of two adjacent rows or columns of a determinant changes the sign of every one of its cofactors.^ Accordingly the adjoint of D^, which we will call B'p may be obtained from D' by rearranging its rows and columns in the same way as the rows and columns of D were rearranged to give JD^ and then prefixing the factor (— 1)" to each element. Let us" now apply the special case already established of our theorem to the determinant B^ and its adjoint D[, the m-rowed minors M^ and M^ being those which are situated in the upper left- hand corner of B^ and I)[ respectively. We thus get (4) J!fi=2)f-» [alg. compl. of M^]. y X FEW PROPERTIES OF DETERMINANTS 33 Now, since Mi is a principal minor, its algebraic complement is the same as its ordinary complement, I and this in turn is the same as the ordinary complement of the minor Min D. Accordingly, using (2), we may write (5) alg. compl. of ilifi = (— 1)" [alg. compl. of If]. Since the elements of M\ differ from those of M' only in having the factor (— 1)" prefixed to each, it follows thatj> (6) ifi = (-l)»"'if'. > We may now reduce (4) by means of (3), (5), and (6).'., We thus get (- lyM' = (- l)»f"'-i)i)»'-i(- 1)" [alg. compl. of Jf ].> Cancelling out the factor ( — 1)™ from both sides of this equation, we see that our theorem is proved. \ We proceed now to point out a number of special cases of this theorem which are worth noting on account of their frequent occur- rence. \ Corollary 1.\ If a^ is any element of a determinant D of the nth order, and if a^ is the cof actor of the corresponding element Aij in the adjoint of D, then _ T^„_2„ This is merely the special case of our general theorem in which m = w — 1, modified, however, slightly in statement by the use of the cof actor «„■ in place of the (w — l)-rowed minor (- Vf^'a-ij. Corollary 2. If J) is any determinant of the nth order and B' its adjoint, then j., _ t^-i This is the special case m = n. Corollary 3. If D is any determinant, and S is the second minor obtained from it by striking out its ith and kth rows and its jth and Ith columns, and if we denote by A^j the cofactor of the element which stands in the ith row and the jth column of D, then This is the special case m = 2.j^ = ( _ Vf+i+^^^DS. \^ /f^ CHAPTER III THE THEORY OF LINEAR DEPENDENCE 12. Definitions and Preliminary Theorems. Two sets of con- stants (flj, Sj, Cj, d,^.,) and (a^, ij, c^, d^) are usually said to be propor- tional to one another if every element of one set may be obtained from the corresponding element of the other by multiplying by the same constant factor. For example, (1, 2, 3, 4) and (2, 4, 6, 8) are proportional. It is ordinarily assumed that either set may be thus obtained from the other, and in most cases this is true ; but in the case of the two sets (1, 2, 3, 4) and (0, 0, 0, 0) we can pass from the first to the second by multiplying by 0, but we cannot pass from the second to the first. A more convenient definition, for many purposes, and one which is easily seen to be equivalent to the above-mentioned one, is the following : Definition 1. The two sets of constants r" -»•" ... t" 1' 2' »' are said to he proportional to each other if two constants c^ and c^, not both zero, exist such that eix[-i-O2x'^ = (£= 1, 2, .■• n). If <?i # 0, we have and if (?2^0, we have The two sets of constants ^t ~i \i ^2' ■■■ ^ni 0, 0, ... 0, THE THEORY OF LINEAR DEPENDENCE 35 are evidently proportional, since if we take Cj = and c, = any con- stant not zero, we have a pair of e's which fulfill the requirements of our definition. Linear dependence may be regarded as a generalization of the conception of proportionality. Instead of two sets of constants we now consider m sets, and give the following : Definition 2. The m sets of n constants each, 4-1, 4'), ... a;W (i = 1, 2, ... wi), are said to he linearly dependent ifm constants o^, c^, ••• o^, not all zero, exist such that c,x^ + c^'j + • . . + e^o; W = 0' = 1, 2, . . . n). If this is not the case, the sets of quantities are said to he linearly independent. In the same way we generalize the familiar conception of the proportionality of two polynomials as follows : Definition 3. The m polynomials {in any numher of independent variahles') f-^, f^, ■■■ f^ are said to he linearly dependent if m constants "jj, Cg, • • • c„, not all zero, exist such that ^l/l + ^2/2 + - + <'mfm=^- If this is not the case, the polynomials are said to he linearly inde- pendent.* The following theorems about linear dependence, while almost self-evident, are of sufficient importance to deserve explicit state- ment : Theorem 1. If m sets of constants {or if m polynomials) are lin- early dependent,it is always possible to express one — hut not necessarily any one — of them linearly in terms of the others. This set of constants {or this polynomial) is then said to he linearly dependent on the others. This is seen at once if we remember that at least one of the e's is not zero. The relations (or relation) in which the e's occur can, then, be divided through by this e. * We might clearly go farther and consider the linear dependence of m sets of r. polynomials each. The two cases of the text would be merely special cases from thi« general point of view. 36 INTRODUCTION TO HIGHER ALGEBRA Theorem 2. If there exist among the sets of constants {or among tht polynomials) a smallernumber of sets {or of polynomials) which are linearly dependent, then the m sets {or the m polynomials) are linearly dependent For suppose there are I sets of constants (or I polynomials) which are linearly dependent {l<m), then we may take for our set of m c's, the I c's which must exist for the I sets (or polynomials) and (ot — l) zeros Theorem 3. If any one of the m sets of constants consists exclu sively of zeros {or if any one of the polynomials is identically zero), the m sets {or the m polynomials) are linearly dependent. For we . may take for the c corresponding to Chis particular set (or polynomial) any constant what(rver, except lero,, and for the othei (m — 1) c's, {m — 1) zeros. 13. The Condition for Linear Dependence^ of t$ets of Constants. In considering m sets of n constants each, (1) 7^lx%...x^^ (i=l, 2, ...m> it will be convenient to distinguish between the two cases m^n and m>n. {a) m^n. We wish here to prove the following fundamental theorem : Theorem 1. A necessary and sufficient condition for the linear dependence of the m sets (1) of n constants each, when m-^n, is that all the m-rowed determinants of the matrix x{ x', . •• < ^I' x'> . ■■ < • • ' ' ■ should vanish. a;W 4"' • ■• 4"' That this is a necessary condition is at once obvious; for if the m sets of constants are linearly dependent, one of the rows can be expressed as a linear combination of the others. Accordingly if in any of the m-rowed determinants we subtract from the elements of this row the corresponding elements of the other rows after each row THE THEORY OF LINEAR DEPENDENCE 37 has been mtiltiplied by a suitable constant, the elements of this row will reduce to zero. The determinant therefore vanishes. We come now to the proof that the vanishing of these deter- minants is also a sufficient condition. We assume, therefore, that all the TO-rowed determinants of the above matrix vanish. Let us also assume that the rank of the matrix is r>0* (of. Definition 3, § 7). Without any real loss of generality we may (and will) assume that the r-rowed determinant which stands in'the upper left-hand corner of the matrix does not vanish ; for by changing the order of the sets of constants and the order of the constants in each set (and these orders are clearly quite immaterial) we can bring one of the non- vanishing y-rowed determinants into this position. We will now prove that the first {r -f- 1) sets of constants are lin- early dependent. From this the linear dependence of the m sets follows by Theorem 2, § 12. Let us denote by Cj, Cji "• ^r+i the cofactors in the (>• -f l)-rowed determinant which stands in the upper left-hand corner of the matrix, and which correspond to the elements of its last column. If we remem- ber that all the (r 4-1)- rowed determinants vanish, we get the relations c^x] -\-c^x'l-\- ■■■ ^ c^+ia;'r'' = (y = »• -f 1, r -I- 2, ■ • • w). Since the sum of the products of the elements of any column of a determinant by the cofactors of the corresponding elements of another column is zero, this equation is also true when/ = 1, 2, ••• r. This establishes the linear dependence of the first (r + V) sets of constants, since c^+i, being the r-rowed determinant which stands in the upper left-hand corner of the matrix, is not zero. (J) m>n. This case can be reduced to the one already considered by the following simple device. Add to each set of n constants m—n zeros. We then have m sets of m constants each. Their matrix con- tains only one w-rowed determinant, and this vanishes since one, at least, of its columns is composed of zeros. Therefore these m sets of m constants each are linearly dependent ; and hence the original m sets of n constants each were linearly dependent. Thus we get the theorem : Thboeem 2. m sets of n constants each are always linearly depen- dent if m-^^n. * In general we shall have r- = m — 1 , but r may have any value less than m. The only case which we here exclude is that in which all the elements of the matrix are zero, a case in which the linear dependence is at once obvious- 88 INTRODUCTION TO HIGHER ALGEBRA EXERCISES Determine whether the following sets of constants are linearly dependent or not : S. 3 a, -2 J, -3c, 6d, a, 0, - c. id, 0, -i. 0, -Zd. 1, 0, 0, 5, 1, 2, 6, 7, 3, 1, 3, 16. 5, 2,.l, 3, 4, 0, 3, 0, 0, 8, 15, 7, 3, 9, 7. 5, -7, 0, 1, -1, 1, -3, -2, 3, -1, I 4, 0, 7, -9, 2. 14. The Linear Dependence of Polynomials. Suppose we have m polynomials, f^j^^ ...f^^ in any number of independent variables. A necessary and sufficient condition for the linear dependence of these polynomials is evidently the linear dependence of their m sets of coefficients. Thus the condi- tions deduced in the last section can be- applied at once to the case of polynomials. EXERCISES Determine whether the following polynomials are linearly depend- ent or not: r ^R^ j^ qn r Id a; + 30 s, 1. J 6a; + 2y + 53- 4, [ 15 a; + 9y -18. 2. 3. 3 Sj + 4 ij — 4 a;, + 6 a;^, 7a;i +Zx^ + 'J x^, 5 Xj + 9 Xj — Xj + 4 X4 + 8. 2a;2+ 8a;y+ 6y^ ■\-\i:X + 12y - 7i» + 3/2+ 6 J. _ 4y^ 33;"- 6x3/+ 8y2- 5x + 7, Ba" + 20 xy + 15 3/2 + 35 x + 30 3/ - 10. THE THEORY OF LINEAR DEPENDENCE 39 15. Geometric Illustrations. The sets of n constants with which we had to deal in §§ 12, 13 may, provided that not all the constants in any one set are zero, advantageously be regarded as the homoge- neous coordinates of points in space of w — 1 dimensions. It will then be convenient to speak of the linear dependence or independ- ence of these points. The geometric meaning of linear dependence will be at once evident from the following theorems for the case w = 4. Two points will here be represented by two sets of four constants ®''°^' ^v 1/v "v tv *2' 2^2' ^2' 2' which will be linearly dependent when, and only when, they are pro- portional, that is, when the points coincide. Hence: Theorem 1. Two points are linearly dependent when, and only when, they coincide. If we have three points in space, P^, P^, Pg, whose coordinates are (zj, y^, z^, t^), {x^, y^, z^, t^), {x^, y^, Zg, t^), respectively, and which are linearly dependent, there must exist three constants Cj, c^ Cj, not all zero, such that OlVl + ^2^2 + ^^3^3 = 0' CiZi + C^Z^ + CgZs = 0, ^1*1 + <?2*2 + ^3^3 = 0- Let us suppose the order of the points to be so taken that Cg gtO, and Xg ^ /Cy^Xj^-Jr rC^X^, ^3 = "-1 ^1 "t" "-2^2' tg = K-^t-^ + n^2 2> solve for Xg, y^, Sg, t^ (1) where Tc^= — cje^, h^= — cjcy Now if Ax-\-By^Qz + I)t = Q is the equation of any plane through the points Pj and Pj, we have Ax^^-By^^ Gz^ + Dt^=0, Ax^ + By^ + Cz^ + Bt^ = 0. 40 INTRODUCTION TO HIGHER ALGEBRA Multiplying the first of these equations by Je^, the second by ^3, and adding, we have, by means of the equations (1), Ax^ + %3 + (7^3 + 2)^3 = 0. Hence every plane through Pj and P^ passes through P3 also, and the three points are collinear. Now, in order to prove conversely that any three collinear points are linearly dependent, let us suppose the three points Pj, P3, Pg collinear. We may assume that these three points are distinct, as otherwise their linear dependence would follow from Theorem 1. We have seen that when three points are linearly dependent, the line through two 'of them contains the third. Hence if we let tV — Ih-t X-t ~\~ '^0*^0? y' = hy\ + ^-iHv • ^ ^~ 11 "I" 2 2' where Aj and k^ are two constants, not both zero, the point (a;', y\ s', <') or P' lies on the line PxP^-, and our theorem will be established if we can show that the constants Tc-^ and h^ can be so chosen that the points P' and Pg coincide. Now let ax -\-hy -\- cz -\- dt = be the equation of any plane through the point P3 but not through Pj or P^. Thus P3 is determined as the intersection of this plane with tlie line PjPg, so that if P', which we know lies on P^Pi.^ can be made to lie in this plane, it must coincide with Pg and the proof is complete. The condition for P' to lie in this plane is ax' + hy' + cz' + dt' = 0. Substituting for cb', y', s', t' their values given above, we have h-lax.^ + %! + cz^ + dt-^ + li^^ax^ + %2 + '^^2. + ^^%} = ^• But neither of these parentheses is zero, since the plane does not pass through Pj or Pg, hence we may give to \ and k^ values different from zero for which this equation is satisfied. We have thus proved Theorem 2. Three points are linearly dependent when, and only when, they are collinear. The proofs of the following theorems are left to be supplied by the reader. It will be found that some of them are readily proved THE THEORY OF LINEAR DEPENDENCE 41 from the definition of linear dependence, as above, while for others it is more convenient to use the condition for linear dependence ob- tained in § 13. Theoeem 3. Four points are linearly dependent when, and only when, they are complanar. Theorem 4. Five_^ or more points are always linearly dependent. Another geometric, application is suggested by the following con- siderations : A set of n ordinary * quantities is nothing more nor less than a complex quantity with n components (of. § 21). Our first definition of linear dependence is therefore precisely equivalent to the following: The m complex quantities J- J- n ft , ,. fi are said to he linearly dependent if m ordinary quantities Cj, c^, ••• e^, not all zero, exist such that : c^a-^ + o^a^+ ■■■ +e^a„ = 0. Now the sim'plest geometric interpretation for a complex quantity with n components is as a vector in space of n dimensions,! and. we are thus led to the conception of linear dependence of vectors. The geometric meaning of this linear _ dependence will be seen from the following theorems for the case w = 3: Theorem 5. Two vectors are linearly dependent when, and only when, they are collinear. Theorem 6. Three vectors are linearly dependent when, and only when, they are complanar. Theorem 7. Four or more vectors are always linearly dependent. In order to get a geometric interpretation of the linear dependence of polynomials, we must consider, not the polynomials themselves, but the equations obtained by equating them to zero. We speak of these equations as being linearly dependent if the polynomials are * Two different standpoints are here possible according as we understand the term ordinary quantity to mean real quantity, or ordinani complex quantity. t There are of course other possible geometric interpretations. Thus in the case M= 4 we may regard our complex quantities as quaternions, and consider the meaning of linear dependence of two, three, or four quaternions. 42 INTRODUCTION TO HIGHER ALGEBRA linearly dependent. If then we regard the independent variables as rectangular coordinates, these equations give us geometric loci in space of as many dimensions as there are independent variables. Thus, in the cases of two and three variables, we have plane curves and surfaces respectively. The case of two loci is of no interest, as they must coincide in order to be linearly dependent. In the case of three linearly dependent loci it is easily shown that any one must meet the other two in all their common points and in no others. The following theorems will serve to illustrate the geometric mean- ing of linear dependence : (1) In the plane : Theorem 8. Three circles are linearly dependent when, and only when, they belong to the same coaxial family . Theorem 9. Four circles are linearly dependent when, and only when, they have a (real or imaginary) common orthogonal circle. Theorem 10. Four circles are linearly dependent when, and only when, the points of intersection of the first and second, q/nd the points of intersection of the third and fourth, lie on a common circle. Theorem 11. Five or more circles are always linearly dependent. (2) In space (using homogeneous coordinates) : Theorem 12. Three planes are linearly dependent when, and only when, they intersect in a line. Theorem 13. Four planes are linearly dependent when, and only when, they intersect in a point. Theorem 14. Five or more planes are always linearly dependent. CHAPTER IV LINEAR EQUATIONS 16. Non-homogeneous Linear Equations. In every elementary treatment of determinants, however brief, it is explained how to solve by determinants a system of n equations of the first degree in n unknowns, provided that the determinant of the coefficients of the unknowns is not zero. Cramer's Rule, by which this is done, is this : Cramer's Rule. Jf in the equations aiiX^-\ h ai„x„ = ki. the determinant «»ia;i+-- + an„a;„= J„ a = *ii *»i »itt is not zero, the equations have one and only one solution, namely : a a^ a where a^ is the n-rowed determinant obtained from a by replacing the elements of the ith column by the elements k-^, k^, •■• /fc„. This rule, whose proof we assume to be known,* is of funda- mental importance in the general theory of linear equations to which we now proceed. * The proof as given in most English and American text-hooks merely establishes the fact that if the equations have a solution it is given by Cramer's formulse. That these formulae really satisfy the equations in all cases is not commonly proved, but may be easily estabished by direct substitution. We leave it for the reader to do this, 43 44 INTRODUCTIOKT TO HIGHER ALGEBRA Consider the system of m linear equations in n variables: aiia;i+ ••• + a i„a;„ + Jj = 0, «».i*i+ ••• + a™a;„+6„ = 0, where m and n may be any positive integers. Three cases arise: (1) The equations may have no solution, in which case they are said to be inconsistent. (2) They may have just one solution. (3) They may have more than one solution, in which case it will pres- ently appear that they necessarily have an infinite number of solutions. Let us consider the two matrices: a = ni *ln *ml b = *ii *ml We will call a the matrix of the system of equations, b the aug- mented matrix. It is evident that the rank of the matrix a cannot be greater than that of the matrix b, since every determinant contained in a is also contained in b. We have, then, two cases: I. Rank of a = Rank of b. II. Rank of a < Rank of b. We will consider Case II first. Let r be the rank of b. Then b must contain at least one r-rowed determinant which is not zero. Moreover, this determinant must contain a column of b's, since otherwise it would be contained in a also, which is contrary to our hypothesis. Suppose for definite- ness that this non-vanishing r-rowed determinant is the one situ- ated in the upper right-hand corner of b. There is no loss of generality in assuming this, since by writing the equations in a dif- ferent order and changing the order of the variables x-^, ■■■ x^ wo can always bring the determinant into this position. Now lot brevity let us represent the polynomials forming the first members of our given equations by V^, F^, ■■■ F^ respectively, and the homogeneous polynomials obtained by omitting the constant terms \n each of these equations by f^, f^, -. f^. Then we have the identities: Fi=fi + b„ (i = l, 2, ... w). LINEAR EQUATIONS 45 Consider the first r of these identities. Since the rank of a is less, than r, the polynomials /i,/2, •••/^ are linearly dependent, '^\S\ + C2/2 + • • • + e«/r = 0, hence c-^F^-V ••■ -^ c^I'^ = e^^-\- ■■■ + c^b^= C. But since the rank of b is r, the polynomials F^, ■■■ F^ are linearly independent and therefore 0^ 0. Hence the given equations are in- consistent, for if they were consistent all the F's would be zero for some suitably chosen values of a;j, ••• a;„, and if we substitute these Values in the last written identity we should have Let us now consider Case I. Let r be the- common rank of a and b, then there is at least one f-rowed determinant in a which is not zero. This same determinant also occurs in b. Suppose it to be situated in the upper left-hand corner of each matrix. Since all (?• + 1) -rowed determinants of either matrix are zero, the first (r-f 1) of the F's are linearly dependent, and we have c^F^ + c^F^ +.-->rO,F,Jf c^+^F,^^ = ; and, since F^, ■■■ F^ are linearly independent, <?^+j cannot be zero ; hence we may divide through by it and express F^+i linearly in terms of Fj^, ■■- F^. The same argument holds if instead of F^^-^ we take ^r+2 o'' ^°y other one of the remaining J"s. Hence F^^i = * ra J-^ + ... + W}F, {1=1,2, ...m- r). From these identities it is obvious that at any point (a;^, ■■■x„) where F-^, ■■■ Fr all vanish, the remaining F's also vanish. In other words, any solution which the first r equations of the given system may have is necessarily a solution of the whole system. Now consider the first r of the given equations. Assign to a^r+n • • • ^n any fixed values 4+i' • • • *»' and transpose all the terms after the rth in each equation to the second member, "11^1 "I •" "■i'-^'- ~ ~ ^1' '•+1^^+! '^i"*" ~ ^1' 46 INTRODUCTION TO HIGHER ALGEBRA Remembering that the right-hand sides of these equations are known constants, and that the determinant of the coefficients on the left is not zero, we see that we have the case to which Cramer's Rule applies, and that this system of equations has therefore just one solution. Hence the given system of equations is consistent, and we have the theorem: Theorem 1. A necessary and sufficient condition for a system of linear equations to be consistent is that the matrix of the system have the same rank as the augmented matrix. From the foregoing considerations we have also Theorem 2. If in a system of linear equations the matrix of the system and the augmented matrix have the same rank r, the values of n — r of the unknowns may he assigned at pleasure and the others will then he uniquely determined. The n—r unknowns whose values may he assigned at pleasure may he chosen in any way provided that the matrix of the coefficients of the remaining unknowns is of rank r. EXERCISES Solve completely the foUowing'systems of equations: 2x- y + Sz- 1 = 0, 4:x-2y- z+ 3 = 0, 2x- y-4:z+ 4 = 0, lOi - 5y - 62 + 10 = 0. 4a:- y+ 2 + 5 = 0, 2x-Zy + 5z + l = 0, x+ y-2z + 2 = Q, 5x — 2+2 = 0. 3. 2x -3y + iz- u) = 3, x + 2y - 2 + 2«) = 1, 3x- 3^ + 22- 3m; =.4, 3a; — y + z — 7w = 4. LINEAR EQUATIONS 47 17. Homogeneous Linear Equations. We will now consider the special case where the equations of the last section are homoffeneout, i.e. where all the 6's are zero, aiiaJiH |-ai„a;„=:0, ami2;i+ •••+«„„«„ = 0. The matrices a and b of the last section differ here only by a column of zeros ; hence thej^ always have the same rank and this is called the rank of the system of equations. Theorems 1 and 2 of the last section become Theorem 1. A system of homogeneous linear equations always has one or more solutions. Theorem 2. If the rank of a system of homogeneous linear equa- tions in n variables is r, the values of n — r of the unknowns may be assigned at pleasure and the others will then be uniquely determined.* If the rank of the equations is n, there witl therefore be only one solution, and this solution is obviously x-^ = X2= ••• = a;„ = 0. Since the rank can never be greater than n, we have Theorem 3. A necessary and sufficient condition for a system of homogeneous linear equations in the n variables {xi, ■ ■ ■ a;„) to have a solu- tion other than ajj = ajg = • • • = a;„ = is that their rank be less than n. Corollary 1. If there are fewer equations than unknowns, the equations always have solutions other than x-^ = X2= ■■■ = x„= 0. Corollary 2. If the number of equations is equal to the number of unknowns, a necessary and sufficient condition for solutions other than x-^ = X2= ■■■ = Xn= is that the determinant of the coefficients be zero. In the special case where the number of equations is just one less than the number of unknowns and the equations are linearly independent, we will prove the following : Theorem 4. Every set of values of x^,---x„ which satisfies a system ofn — 1 linearly independent,^ homogeneous linear equations in * Cf. also the closing lines of Theorem 2, § 16. t The theorem is still true if the equations are linearly dependent, hut it is then trivial, since the determinants in question are all zero. 48 INTRODUCTION TO HIGHER ALGEBRA n unknowns is proportional to the set of (n — Vyrowed determinanti taken alternately with plus and minus signs, and obtained by striking out from the matrix of the coefficients first the first column, then the second, etc. Let us denote by a^ the (n — l)-rowed determinant obtained by- striking out the ith column from the matrix of the equations. Since the equations are linearly independent, there must be at least one of the determinants a^, a^, ■■■ a„ which is not zero. Let it be a^. Now assign to a;,- any iixed value, o, and transpose the I'th term of each equation to the second member and we have Oi^x^ +•■■ + ai,i-ia;i_i + aj.j+i ^i+j + ■■• + «!„ »„ = - a^^c, Hence: Xu = ^~ ' — '-^ i^= 1, 2, ••• m), from which it is clea» that (x^, ■■• «„) are proportional to the de- • terminants (aj, — a^, a^, ... ( _ l)"-!^^), as was to be proved. The theory of homogeneous linear equations has here been de- duced from the theory of linear dependence. It can, however, in turn be used to obtain further results in this last-mentioned theory. As an example of this we will deduce the following theorem, which we shall find useful later : Theorem 5. If a set of points {x-^, ■■■ x„), finite or infinite in number, have the property that k points can be found among them upon which every other point of the set is linearly dependent, then any k + 1 points of the set will be linearly dependent. Let {x[, ■■■ x^), {x>l, ■■■x'l),-- (4*1, ... 4«) be the k points upon which every other point of the set is linearly dependent, and let (Xi, ... X'„), (X'l, ... X'J:), ... (X^ii, ... Xt*H-i]) be any k+1 points of the set. Then we may write ■X M = cW x[ + dl 44-... + cfx['^ , (1) ■ ■ ■ ' .'..'', ■ (»■ = !> 2, ...^-H) .X^ = c^fx'„ + 4*14' -f- ... + cfx^. LINEAR EtiUATIONS 49 This is true by hypothesis if {Xf, ••• Xi^) is not one of the first A points, and if it is one of these points, it is obviously true. We have then to prove that k+l constants, Cj, 0^, ••• C^+p not all zero, can be found such that C,X^+C,X'i+... + O,,,X}''-^^ = (y=i, 2, ...«). By substituting here the values of the X's from (1), we see that these equations will be fulfilled if Cici+C^c'^+.:+C,^,c, [k+l] . 0, and this is a system of fewer equations than unknowns, which is therefore satisfied by a set of C's not all zero. (Cf. Theorem 3. Cor. 1.) IIXERCISES Solve completely the following systems of equations ; 1. 2. lla;+ Sy-2z+ 3w=0, 2x+ Sy- 2+ 2w = 0, 7 X— y+ z - 3k' = 0, 4:x-lly+5z- 12w = 0. 2x~ Sy + 5z+ 3w = 0, ix — y + 2+ ro = 0, Sx- 2y + Sz+ 4to = 0. 18. Fundamental Systems of Solutions of Homogeneous Linear Equations. If {x[, ■ ■ -x'^) is a solution of the system of equations (1) nl"^! I" ■ ■ ■ "r ^mn-^B — "» then {ex[,'--cxn) is also a solution, and by giving to e different values we get thus (except in the special case in which the a;"s are all zero) an infinite number of solutions. These may include all the solutions of (1) (cf. Theorem 4 of the last section), but in general this will not be the case. 50 INTRODUCTION TO HIGHER ALGEBRA Suppose, again, that (x[, ■■■ xl) and (x'^, ■■■ x'l) are two solutions ol (1), then {c^Xj + c^x'[, ■ ■ ■ c^x^ + e^x'^ is also a solution. If the two given solutions are proportional to each other, this clearly gives us nothing more than what we had ahove by starting from a single solution ; but if these two solutions are linearly independent, we build up from them, by allowing c^ and c^ to take on all values, a doubly infinite system of solutions ; but even this system will usually not include all the solutions of (1). Similarly we see that, if we can find three linearly independent solutions, we can build up from them a triply infinite system of solutions, etc. If, proceeding in this way, we succeed in finding a finite number of linearly independent solutions in terms of which all solutions can be expressed, this finite number of solutions is said to form a fundamental system. Definition. If {x^^, ■■■ x^'S) (i= 1, 2, ••• ^) are a system of k solutions of {!) which satisfy the following two conditions, they are said to form a fundamental system : (a) They shall be linearly independent. (b) Every solution of (\) shall he expressible in the form {c^x[ + c^x'l -t- • • • + e^^l\ c^x'„ + c^xl+- + c;^[*l ). Theorem 1. If the equations (1) are of rank r <n, they possess an infinite number of fundamental systems each of which consists of n — r solutions. Suppose the r-rowed determinant which stands in the upper left- hand corner of the matrix of the equations (1) does not vanish, and let us consider the first r of these equations. Any solution of these will be a solution of all the others. Transpose all terms after the rth to the second members, and let (a^^+j, ••■ a;„) have any fixed set of values (a;J.+j, ••• x^), not all zero; then these r equations will have just one solution given by Cramer's Rule. Call it (a/j, ••■ x'„). Now let (a;r+i, ■■■ x^ have any other fixed set of values {xJl-^, ■■■ a/^), not all zero, and we get another solution, (a;'/, ... x'j^. Continue in this way until we have n—r solutions X-y, ••■ X^, ^r+v ••• ^ni a^i, ... 4»-^l,a;[;!-i, ••■ 4»^l LINEAR EQUATIONS 51 If we have chosen these n — r sets of values for (x^ the determinant (2) 'r+l' »„) SO that "r+l 4+T^"-4""''^ is not zero, — and this may clearly be done in an infinite variety of ways, — these n — r solutions will be linearly independent. That ia to say, we may thus obtain an infinite number of sets of w — r solu. tions each, each of which satisfies condition (a) of our definition foj a fundamental system. To prove that these sets of solutions also satisfy condition (J), let us suppose that (Xj, ■ • • X„) is any solution of the r equations we are considering. The last n— r oi these -X^s are linearly dependent on the n—r sets of values we have chosen for (aj^+i, ■ • • a;„) since we have here more sets of constants than there are elements in each set (cf. Theorem 2, §13), and the determinant (2) is not zero. Thus (3) Xi = c^x[+c^x>l + --^o,_^f-'-^ (i = r+l, r4-2, - n). Let us now solve the first r equations (1) by Cramer's Rule, regard- ing a;^+j, •■■ x„ as known. We thus get results of the form ,- = ^: Lj-a;^+2 + A'!x,^^+-+A^^^x^ By assigning special values here to x. •r+l' 0' = 1,2, ^v^ ^e get r). (4) Xj = J^.X,+i + A'!X,^^ +■■■+ Af-'-^X,. (y = l,2, ...r) If we multiply the first n — r of these equations by Cj, ••• e^ spectively and add, we get, by (3), Ci4 + ■ • ■ + c^-A"~'^ = ^j^r+i + ■ • • + 4""'''-^«- Consequently, by the last equation (4), (5) , Xj = c,x^ +■■■ c„_,4"-'' 0" = 1, 2, • Equations (8) and (5) together prove our theorem. , re- r\ 52 INTRODUCTION TO HIGHER ALGEBRA We thus see that the totality of all solutions of the system (Ij forms a set of points satisfying the conditions of Theorem 5, §17. Consequently, ^ Theorem 2. If the rank of a system of homogeneous linear equa- tions in n variables is r, then any w — r + 1 solutions are linearly dependent. Finally we will prove the theorem. Theorem 3. A necessary and sufficient condition that a set of solutions of a system of homogeneous linear equations of rank r in n variables form a fundamental system is that they be (a) linearly independent, (5) n — r in number. By definition, (a) is a necessary condition. To see that (5) also is necessary, notice that by Theorem 2 there cannot be more than n — r linearly independent solutions. We have, then, merely to show that I linearly independent solutions never form a fundamental system when l< n — r. If they did, then by Theorem 5, § 17, any set oi I -\- 1 solutions would be linearly dependent, and therefore the same would be true of any set of n — r solutions (since n — r^l + 1). But by Theorem 1, this is not true. In order now to prove that conditions (a) and (6) are also sufficient, let {xf, xf, ... rcW) {i = l,2,...n-r) be any system oi n — r linearly independent solutions of our system of equations, and let {x^, ... x„) be any solution of the system. Then, by Theorem 2, we have n — r +1 constants (cj, ... 'c„-r+i)i not all zero, and such that i <?ia;j + c^x'! + ... + o^.^xf-''^ + c„.^^^Zj ~0 (/ = 1, 2, ... n). But since the n — r given points are linearly independent, c„_^^.j =^ 0; accordingly these last equations enable us to express the solu- tion (xi, ... x^) linearly in terms of the n — r given solutions, and this shows that these n — r solutions form a fundamental system. LINEAR EQUATIONS 63 EXERCISES 1. Prove that all the fundamental systems of solutions of a system of homo- geneous linear equations are included in the infinite number obtained in the proof of Theorem 1. 2. Given three planes in space by their equations in homogeneous coordinates. What are their relative positions when the rank of the system of equations is 3? when it is 2 ? when it is 1 ? 3. Given three planes in space by their equations in non-homogeneous coordi- nates. What are their relative positions for the different possible pairs of values of the ranks of the matrices and augmented matrices? CHAPTER V SOME THEOREMS CONCERNING THE RANK OF A MATRIX 19. General Matrices. In order to show that a given matrix is of rank r, we have first to show that at least one r-rowed deter- minant of the matrix is not zero, and secondly that all {r + 1)- towed determinants are zero. This latter work may be considerably shortened by the following theorem : Theorem 1. If in a given matrix a certain r-rowed determinant is not zero, and all the {r+l)-rowed determinants of which this r-rowed determinant is a first minor are zero, then all the (r-'r'\.)-rowed deter- minants of the matrix are zero. We will assume, as we may do without loss of generality, that the non-vanishing r-rowed determinant stands in the upper left- hand corner of the matrix. Let the matrix be nr •*ml *ln and consider the r -f 1 sets of n quantities each which lie in the first r-\-l rows of this matrix. These r 4- 1 sets of quantities are linearly dependent, as will be seen by reference to the proof of Theorem 1, § 13, for although we knew there that all the (r -1- l)-rowed deter- minants were zero, we made use of this fact only for those {r-\-V)- rowed determinants which we now assume to be zero. Moreover, since the r sets of constants which stand in the first r rows of our matrix are linearly independent, it follows that the (r -|- l)th row is linearly dependent on the first r. Precisely the same reasoning shows that each of the subsequent rows is linearly dependent on the first >• rows. Accordingly, by Theorem 5, §17, any r-\-\ rows are linearly dependent ; and therefore, by Theorem 1, §13, all the (r-f-l)- rowed determinants of our matrix are zero, as was to be proved. 64 THEOREMS CONCERNING THE RANK OP A MATRIX 56 Still another method of facilitating the determination of the rank of a matrix is by changing the form of the matrix in certain ways which do not change its rank. In order to explain this method, we begin by laying down the following definition : , Definition 1. By an elementary transformation of a matrix we understand a transformation of any one of the following forms : (a) the interchange of two rows or of two columns ; [b) the multiplication of each element of a row {or column) by the same constant not zero ; iji) the addition to the elements of one row {or column) of the prod- ucts of the corresponding elements of another row {or column) by one and the same constant. It is clear that if we can pass from a matrix a to a matrix b by one of these transformations, we can pass back from b to a by an elemen- tary transformation. Definition 2. Two matrices are said to he equivalent if it is possi- ble to pass from one to the other hy a finite number of elementary trans- formations. Theorem 2. If two matrices are equivalent, they have the same rank. It is evident that the transformations {a) and (5) of Definition 1 do not change the rank of a matrix, since they do not affect the van- ishing or non-vanishing of any determinant of the matrix. In order to prove our theorem, it is therefore sufficient to prove that the rank of a matrix is not changed by a transformation (e). Suppose this transformation consists in adding to the elements of the pth. row of a matrix a k times the elements of the qih row, thus giving the matrix b. Let r be the rank of the matrix a. We will first show that this rank cannot be increased by the transforma- tion, that is, that all {r + l)-rowed determinants of the matrix b are zero. By hypothesis all the (?• -f 1 )-rowed determinants of the matrix a are zero, and some of these determinants are clearly not changed by the transformation, namely, those which do not contain the j9th row, or which contain both the pth. and the qt\\ row. The other determinants, which contain the pth row but not the g'th, take on after the transformation the form ^± A-B where A and B are {r -I- l)-rowed determinants of a, and are therefore zero. Thus we see that the transformation (e) never increases the rank of a matrix. 66 INTRODUCTION TO HIGHER ALGEBRA Moreover, the rank of b cannot be less than that of a, for then the transformation (c) which carries b into a would increase the rank of b, and this we have just seen is impossible. This theorem can often be used to advantage in determining the rank of a matrix, for by means of elementary transformations it is often easy to simplify the matrix very materially. BXERCISES Determine the ranks of the following matrices: 14 12 6 8 2 6 104 21 9 17 7 6 3 4 1 35 30 15 20 5 75 116 - 39 • 171 - 69 402 123 45 301 87 -417 -169 1 14 -46 268 82 30 3. Prove that any matrix of rank r can be reduced by means of elementaxy transformations to a form where the element in the ith row and ith column is 1 when i<r, while all the other elements of the matrix are zero. 4. Hence prove that two matrices with m rows and n columns each are always equivalent when they have the same rank. 5. Prove that a necessary and sufficient condition that the matrix be of rank or 1 is that there exist m + n constants oi, ••• Om, Pi, ••• j3n such that 20. Symmetrical Matrices. Definition. The square matrix «u «12 •■ •«1« «21 «22" •«2™ a = • • • • ■ • • «.l ««2 «™ '^and also its determinant) is said to he symmetrical if the pairs of termi which are situated symmetrically with respect to the principal diagonal are equal. That is, if a^y = %. THEOREMS CONCERNING THE RANK OF A MATRIX 57 We will denote by Mf an i-rowed principal minor of a. It is onr main object in this section to show how the rank of the symmet- rical matrix may be determined by an examination of the principal minors only. This may be done by means of the following three theorems. Theorem 1. ff an r-rowed principal minor Mr of the symmetrical matrix a is not zero, while all the principal minors obtained hy adding one row and the- same column, and also all those obtained by adding two rows and the same two columns, to M^ are zero, then the rank of a is r. Let the non- vanishing minor be the one which stands in the upper left-hand corner of a, and let B^^ denote the determinant obtained by adding the ath row and the j8th column to M^. If we can show that B^a = for all unequal values of « and /S our theorem will be proved. Cf. Theorem 1, § 19. Give to the integers « and ^ any two unequal values, and let C denote the determinant obtained by adding to M^ the ath and ySth rows and the «th and /3th columns of a. Then we have, . by hypothesis, M^i^ 0, B^ = 0, B^^ = 0, C= 0. Let M^ be the two- rowed principal minor of the adjoint of Q which corresponds to the complement of Mr in O. Then by Corollary 3, § 11, we have M^=OMr=0. But Mi~B^^Bp^-B^. Therefore -B„8 = 0. Theorem 2. J^ all the (r+l)-rowed principal minors of the sym- metrical matrix a are zero, and also all the {r + 2) -rowed principal minors, then the rank of a is r or less. lir=0, all the elements in the principal diagonal are zero and aU ■ the two-rowed principal minors are zero. That is, an ■ Ujj — a |- = 0, and therefore, since a« = %=0, aij = 0. That is, every element is zero and hence the rank is zero, and the theorem is true in this special case. > Now, assume it true when r = k; that is, we assume that when all {k-\-l)-TOwed principal minors are zero and all (A-|-2)-rowed principal minors are zero, the rank of a is less than k+ 1. Then it follows that when all (yfc-h2)-rowed, and all (A-t- 3)-rowed principal minors are zero. 58 INTRODUCTION TO HIGHER ALGEBRA the rank of a is less than k + 2. For in this case, if all (Jc + l)-rowed principal minors are zero, the rank is less than ^ + 1, by hypothesis, and if some (^ + l)-rowed principal minor is not zero, the rank is ex- actly A + 1, by the last theorem. We see then that if the theorem is true for r = k it is true for r=k + l. But we have proved it true for r = 0, hence it is true for all values of r. Theorem 3. If the rank of the symmetrical matrix a is r > 0, there is at least one r-rowed principal minor of a which is not zero. For all (r+l)-rowed principal minors are zero, and, if all r-rowed principal minors were zero also, the rank of a would be r— 1 or less, by the last theorem. We close with a theorem of a somewhat special character which will be found useful later (cf. Exercises 4-6, § 50). Theorem 4. If the rank of the symmetrical matrix a is r> 0, we may shift the rows {at the same time shifting the columns in the same way, thus keeping a symmetrical) in such a way that no consecutive two of the set of quantities ]y[ ^ ^ ... M shall he zero and M^ =?t ; M^ being unity, and the other M^s being the principal minors of a of orders indicated by their subscripts, which stand in the upper left-hand corner of a after the shifting. By definition we have il!f^=?fcO. Leaving aside for the moment the special case in which all the elements of the principal diagonal are zero, let us suppose the element a^ is not zero. Then by shifting the ith row and column to the first place, we have ilfj ^ 0. We have tn'as fixed the first row and column, but we are still at liberty to shift all the others. Now consider the two-rowed principal minor obtained by adding to M^ one row and the same column. Leaving aside still the special case in which these are all zero, let us suppose that the two-rowed determinant obtained by striking out all the rows' and columns except those numbered 1 and i^ is not zero. Then, by shifting the ijth row and column into the second place, we have M^ "^ ^' W® ii^^t have to consider the three-rowed principal minors of which M^ is a first minor. We can evidently proceed in this way until we have so shifted our rows and columns that none of the quan- tities Mq, ilfj, ... M^ are zero, unless at a certain stage we find that all the principal minors of a certain order which we have to consider are zero. In this case we should have so shifted our first k rows and columns that none of the quantities Mq, M\, ... Mh are zero, but we THEOREMS CONCERNING THE RANK OF A MATRIX 59 should then find that all (A;+l)-rowed principal minors of which M^ is a first minor vanish, so that, however we may shift the last n — Jc rows and columns, we have -M^+j = 0. Let us then examine the (A+2)-rowed principal minors of which M^ is a second minor.* These can (by Theorem 1) not all be zero as otherwise the rank of a would he k <r. That is, if M/i+i = 0, we can so arrange the rows and columns that Mn^^^O. Thus we see that the rows and col- umns of a may be so shifted that no consecutive two of the iKTs are zero. Now, if Mr_i=0, the above proof shows that we can make M^ =^ 0. But even though illf^_ j =?;= we can still make M^ ^ 0, for by hypothesis f all the determinants obtained by adding to M^-i two rows and the same two columns vanish, and if all those obtained by adding one row and the same column were zero also, the rank of a would be ? — 1, by Theorem 1. A symmetrical matrix is said to be arranged in normal form when no consecutive two of the M's of Theorem 4 are zero and Mr=^0. EXERCISES 1. Determine the ranks of the following matrices : 2 1 11 2 4 10 1 1 4 -1 4 8 18 7 f 11 4 56 5 10 18 40 17 2 -1 5 -6 1 7 17 3 1 1 4 1 b d 1 2 5 1 c e 1 3 6 J b c 2bc cd+be 1 2 3 14 32 d e cd+be 2de 4 5 6 32 77 2. By a skew-symmetric determinant, or matrix, is meant one in which Otyw — flj-i (and therefore an = 0). Establish for such matrices theorems similar to Theorems 1, 2, 3 of this section. 3. By considering the effect of changing rows into columns, prove that a skew- gymmetric determinant of odd order is always zero. 4. Prove that the rank of a skew-symmetric matrix is always even. * The tacit assumption is here made that when k = r-l,r<n,aa otherwise Mi+, lyould have no meaning. The case r = n can, however, obviously not occur here, for then we should have itf* + 1 = o ^t 0. t Here again we assume that r < n, for if r = n, Mr = a=^0. CHAPTER VI LINEAR TRANSFORMATIONS AND THE COMBINATION OF MATRICES 21. Matrices as Complex Quantities. "We have said in § 7 that a matrix of m rows and n columns is not a quantity, but a set of mn quantities. This statement is true only if we restrict the term quantity to the real and complex quantities of ordinary algebra. A moment's reflection, however, will show that the conception of quan- tity as used in arithmetic and algebra has been gradually enlarged from the primitive conception of the positive integer by using tha word quantity to denote entities which, at an earlier stage, would not have been regarded as quantities at all, as, for instance, nega- tive quantities. We will consider here only one of these extensions, namely the introduction of complex quantities, as this will lead us to look at our matrices from a broader point of view. If we have objects of two or more different kinds which can be counted or measured, and if we consider aggregates of such objects, we get concrete examples of complex quantities, as, for instance, 5 horses, 3 cows, and 7 sheep. A convenient way to write such a complex quantity is (5, 3, 7), it being agreed that, in the illustra- tion we are considering, the first place shall .always indicate horses, the second cows, and the third sheep. In the abstract theory of complex quantities we do not specify any concrete objects such as horses, cows, etc., but merely consider sets of quantities (couples, triplets, etc.), distinguishing these quantities by the position they occupy in our symbol. Such a complex quantity we often find it convenient to designate by a single letter, a. = (a, b, o) just as in ordinary algebra we denote a fraction (| for instance), which really involves two numbers, by a single letter. We speak here of the simple quantities a, I, c of which a is composed as its first, 60 LINEAR TRANSFORMATIONS AND MATRICES 61 (Second, third components ; and we call two complex quantities equal when and only when the components of one are equal respectively to the corresponding components of the other. Similarly a complex quan- tity is said to vanish when and only when all of its components are zero. What makes it worth while to speak of such sets of quantities as complex quantities is that it is found useful to perform certain alge- braic operations on them. By the sum and difference of two complex quantities • a, = {a„b„e,), a, = ia,,h„c,) we mean the two new complex quantities «i + «2 = («i + <*2? ^1 + K "i + ^2)) «! — «2 = (% — C'2-' \ — ^2! H — ''i)* When it comes to the question of defining what we shall under- stand by the product of two complex quantities, things are by no means so simple. It is necessary here to lay down some rule accord- ing to which, when two complex quantities are given, a third, which we call their product, is determined. Such rules may be laid down in an infinite variety of ways, and each such rule gives us a differenb system of complex quantities, f We come now to the subject of matrices. A matrix of m rows and n columns being merely a set of mn quantities (which we assume to be either real quantities or the ordinary complex quantities of elementary algebra) arranged in a definite order, is, according to the point of view we have explained, a complex quantity with mn components; and it is only a special application of the theory of complex quantities which we have sketched, when we lay down the following definitions: Definition 1. A matrix is said to he zero when and only when all of its elements are zero. Definition 2. Two matrices are said to he equal when and only when they have the same number of rows and of columns, and every element of one is equal to the corresponding element of the other. * That this is the natural meaning to be attached to the terms sum and difference will be seen by reference to the concrete illustration given above. t If, in particular, we wish to introduce the ordinary system of complex quanti- ties of elementary algebra, we use a system of couples, and define the product of two couples, «! = (cti, 61). «2 = ('^2! ^2)' by the formula aiWz = (0102 - 6162, 0162 + a^bi). For further details cf. Burkhardt's Funktionentheorie, §§ 2, 3. 62 INTRODUCTION TO HIGHER ALGEBRA Definition 3. By the sum {or difference) of two matrices of m rows and n columns each, we understand a matrix of m rows and n col- umns, each of whose elements is the sum (or difference) of the corre- sponding elements of the given matrices. In order to distinguish them from matrices, we will call the ordinary quantities of algebra (real quantities and ordinary complex quantities) scalars. Before proceeding, as we shall do in the next section, to the definition of the product of two matrices, we will define the product of a matrix and a scalar. Definition 4. If & is a matrix * and k a scalar, then by the prod- uct Tcdi or sJe we understand the matrix each of whose elements is k times the corresponding element of a. As an obvious consequence of our definitions we state the theorem : Theorem. All the laws of ordinary algebra hold for the addition or subtraction of matrices and their multiplication by scalars. For instance, if a, b, c are matrices, and k, I scalars, a + b = b + a, a + (b + c) = (a + b) + c, ka-\-kh = k{& + b), Aa + Za = (A; + Z)a.t EXERCISE If r\ and r^ are the ranks of two matrices and R the rank of their sum, prove that „ ^ R^r^ + r^- 22. The Multiplication of Matrices. Up to this point we have con- sidered matrices with m rows and n columns. For tlie sake . of sim- plicity of statement, we shall confine ouv attention from now on to square matrices, that is to the case m = n. This involves no real loss * The notation here used, matrices being denoted by heavy-faced type, will be systematically followed in this book. t We add that, as a matter of notation, we shall write (-l)a=-a. LINEAR TRANSFORMATIONS AND MATRICES 63 of generality provided we agree to consider a matrix of m rows and n columns, where m^n, as equivalent to a square matrix of order equal to the larger of the two integers m, n and obtained from the given matrix by filling in the lacking rows or columns with zeros. The question now presents itself: How shall we define the prod- uct of two square matrices of the same order ? It must be clearly understood that we are logically free to lay down here such definition as we please, and that the definition we select is preferable to others not on any a priori grounds, but only because it turns out to be more useful. We select the following definition, which is suggested * by the multiplication theorem for determinants : Definition 1. The product ab of two square matrices of the nth order is a square matrix of the nth order in which the element which lies in the ith row and jth column is obtained hy multiplying each element of the ith row of a hy the corresponding element of the jth column of b and adding the results. Let us denote by «;,• and J^y the elements in the ith row and jth. column of a and b respectively, or, as we will say for brevity, the element (i, j) of these matrices. Then, according to our definition, the element (i, j) of the product ab is (1) a ijby + a^aigy + • • ■ + «.A/t while the element (i, j) in the matrix ba is (2) aj/ii + a^jbi^ + • • • + a„^J»„- Since the two quantities (1) and (2) are not in general equal, we obtain Theorem 1. The multiplication of matrices is not in general com- mutative, that is, in general ^^ , |jg^_ Let us now consider a third matrix c whose element (i, j) is c^y, and form the product (ab)c. The element {i, j) of this matrix is (a.jjji + aigSji -I 1- «iAi>i; (3) + • • + («il*ln + «i2*2n + ••• + ".iifinnj^nj- * Historically this deflnition was suggested to Cayley by the consideration of the composition ot linear transformations; cf. § 23. 64 INTRODUCTION TO HIGHER ALGEBRA On the other hand, the element {i, j) of the matrix a(bc) ia ( 4) + ai-ihlHi + ^22^27 + • • • + hn^nj) + Since the two quantities (3) and (4) are equal, we have estahlished Theorem 2. The multiplication of matrices is associative, that is, (ab)c = a(bc). Finally, since the element {i,j) of the matrix a (b + c) is clearly- equal to the sum of the elements {i, j) of the matrices ab and ac, we have the result Theoeem 3. The multiplication of matrices is distributive, that is, a(b + c) = ab + ac. Besides the commutative, associative, and distributive laws, there is one other principle of elementary algebra which is of constant use, namely, the principle that a product cannot vanish unless at least one of the factors is zero. Simple examples show that this is not true in the algebra of matrices. We have, for instance. (5) "21 -^31 -^22 *32 31 32 = 0, whatever the values of the a's and 5's may be. Hence Theorem 4. From the vanishing of the product of two or more matrices, we cannot infer that one of the factors is zero. The process of cancelling out non-vanishing factors which enter throughout an equation will, therefore, be inadmissible in the algebra of matrices. We next state a result which follows at once from the similarity between the theorem for the multiplication of determinants and our definition of the product of two matrices : Theorem 5. The determinant of a matrix which is obtained by multiplying together two or more matrices is equal to the product of th» 'ieferminants of these matrices. LINEAR TRANSFORMATIONS AND MATRICES 65 The conception of the conjugate of a matrix, as defined in § 7, Definition 2, is an important one, and the following theorem concern- ing it is often useful: Thbokem 6. The conjugate of the product of any number of matrices is the product of their conjugates taken in the reverse order. In order to prove this theorem we first notice that its truth in the case of two matrices follows at once from the definition of the prod- uct of two matrices. Its truth will therefore follow in all cases if, assuming the theorem to be true for the product of w — 1 matrices, we can prove that it is true fot the product of n matrices. Let us write = 3.2 33 ■•■ 3»' Then, from what we have assumed, D = &n ■ ■ ■ ^3 32) where we use accents to denote conjugates. Accordingly, (ai32 - a„y = (aib)' = b'ai = ai - 43^, and our theorem is proved. In conclusion we lay down the following : Definition 2. A square matrix is said to he singular if its deter- minant is zero. According to the convention made at the beginning of this sec- tion, it will be seen that all matrices which are not square are singular. EXERCISES 1. Definition. A matrix a is called a divisor of zero if a matrix b different from zero exists such that either ab = or ba = 0. Prove that every matrix one of whose rows or columns is composed wholly of zeros is a divisor of zero. 2. If it is possible to pass from a to b by means of an elementary transformar tion (cf . § 19, Definition 1), prove that there either exists a non-singular matrix c such that ac = b, or a non-singular matrix d such that da = b. p 66 INTRODUCTION TO HIGHER ALGEBRA 3. If aU the elements of a matrix are veal, and if the product of this matrix and its conjugate is zero, prove that the matrix itself is zero. 4. If the corresponding elements of two matrices a and b are conjugate imagi- naries, and, b' being the matrix conjugate to b, if ab' = 0, then a = b = 0. 23. Linear Transformation. Before going farther with the theory of matrices we will take up, in this section and the next, the closely allied subject of linear transformation, which may be regarded as one of the most important applications of the theory of matrices. In algebra and analysis we frequently have occasion to introduce, in place of the unknowns, or variables, we had originally to deal with, certain functions of these quantities which we regard as new unknowns or variables. Such a transformation, or change of vari- ables, is particularly simple, and for many purposes particularly important, if the functions in question are homogeneous linear poly- nomials. It is then called a homogeneous linear transformation, or, as we shall say for brevity, simply a linear transformation. If x-^, ■■■ x„ are the original variables, and x^, ■■■ x'^ the new ones, we have, as the formulae for the transformation, x[ = ai^x^+ ■■■ +a^„x„. , ^n — ^nl^l + • • • + <^nn^» The square matrix a = is called the matrix of the transformation, and the determinant of this matrix, which we will represent by a, is called the determinant of the transformation. Inasmuch as the transformation is com- pletely determined by its matrix, no confusion will arise if we speak of the transformation a. In most cases where we have occasion to use a transformation it is important for us to be able, in the course of our work, to pass back to the original variables, and for this purpose it must be possible, not merely to express x{, ■■■ xl as functions of x^,--- x„, but also to express a^i, ■•■ x„ as functions of x[, ■■■ x\. In the case of linear transforma- LINEAR TRANSFORMATIONS AND MATRICES 6\ tions this can in general be done. For the equations of the transfer mation may be regarded as non-homogeneous linear equations in Xi, ■■■ x„, and if the determinant a of the transformation is not zero, they can be solved and give a a .„ = ^.i + A where A.-,, •■• A„„ are the cof actors of a,,, ■•• a. in a. This transformation A is called the inverse of the transformation a, but it must be remembered that it exists only if a :^ 0. A linear transformation for which a = is called a singular transformation. If a is non-singular, its inverse A is also non-singular, since the deter, minant of A is a~^(ci. Corollary 2, § 11). Definition. The special linear transformation 1 V 2 — 2' » — »' whose matrix is 1 . . 1 . . ■ • • ■ 1 is called the identical transformation. The determinant of this transformation is 1. We turn now to the subject of the composition of linear trans- formations. If we introduce a new set of variables a;' as functions of the original variables x, and then make a second transformation by introducing a third set of variables x" as functions of the vari- ables x', these two transformations can obviouslj"^ be combined and the variables x" expressed directly in terms of the a;'s. If the two transformations which we combine are linear transformations, it ia readily seen that the resulting transformation will also be linear. The precise formulae are important here, and for the sake of simplic- ity we will write them in the case of three variables, a case which will be seen to be perfectly typical of the general case. 68 INTRODUCTION TO HIGHER ALGEBRA Let •^2 ^^ iV"^! ' ^22'*2 ' ^23"''3' Xg = Pgia^l + OgjaJg + Ogg^g, be two linear transformations. Replacing the a;''s in b by theii values from a, we get < 3^1 = (an^ii + a^i^ia + «3i^i3)«i + («12^11 + «'22^2 + «32*13)a^2 + («i30n + '^23"12 + *33"13)^3' 4' = («11^21 + «21^22 + «31*23)^1 + (<*12"21 "I" ^22^22 "I" *32"23)'''2 + ((I13O21 + ^23^22 "I" ^33 23)'^3' ^8 ~ ("'ll^Sl + *21^32 + '*31"33)^1 "I" ('"iz'^Sl "I" *22''32 "I" *32"33)'^2 + (^13^31 + ^23^32 + '''33"33)*3" It will be seen that the matrix of this transformation is ba. Hence, Theorem. If we pass from the variables x to the variables x' by a linear transformation of matrix a, and from the variables x' to the vari- ables x" by another linear transformation of matrix b, then the linear transformation' of matrix ba will carry us directly from the variables x to the variables x" .* ' 24. CoUineation. We come now to an important geometrical application of the subject of linear transformation. For the sake of simplicity we begin with the case of three variables, Which we will regard as the homogeneous coordinates of points in a plane. The equations . ^, _ ^^^ ^ j^^ ^ ^^^^ (1) ■ y' = a^x + b^y + c^t, . t' = a^x + b^y + Cgi * This result may be remembered conveniently by means of the following symbolio notation, which is often convenient. Let us denote the transformation a by the symbolic equation x' = a(a;), and the transformation b by x'' = b(a;'). The result o< combining these two transformations is then x" = b(a(a;)) or simply x" = ba(s) LINEAR TRANSFORMATIONS AND MATRICES 69 may be regarded as defining a transformation of the points of the plane ; that is, if (a;, «/, t) is an arbitrarily given point, we can com- pute, by means of (1), the coordinates {x\ y\ t') of a second point into which we regard the first point as being transformed. The only exception is when the computed values of x\ y\ t' are all three zero, in which case there is no point into which the given point is transformed. This exceptional case can clearly occur only when the determinant of the transformation (1) is zero. Let us then confine our attention to non-singular linear transformations. In this case, not only does every point (a;, y, t) correspond to a dfefinite point («', y\ t'), but conversely, every point (x', y\ t') corresponds to a definite point (a;, y, t), since the transformation (1) now has an inverse (2) D B" B B B" B where B is the determinant of (1), and J.^, Bi^ Q^ are the cofactors 'va.B. The points (a;, y, t) of the line (3) ax + ^y + 'yt=^0 are transformed by means of the non-singular transformation (1) into points of another line, ^^^ aA^ + ^B^ + yC, ^, ^ aA^ + ^B^+yO^ ^, ^ aAs+ 0B^ + yC^ ^, ^^^ as we see by using formulae (2). Conversely every point .of the line (4) corresponds, as we see by using (1), to a point on (3). That is, the transformation establishes a one-to-one correspondence between the points on the two lines (3) and (4), or, as we say, it transforms the line (3) into the line (4). On account of this property of trans- forming straight lines into straight lines, the transformation is called a collineation. The transformation is also known as a projective transformation, for it may be shown that it can be effected by pro- jecting one plane on to another by means of straight lines radiating from a point in space. 70 INTRODUCTION TO HIGHER ALGEBRA What we have here said in the case of two dimensions applies with no essential change to three dimensions. The transformation x' = a-^ + hy^ + CjZ + d-f, y' = a^x + l^y + c^z + d^, z' = a^x + i^ + CgZ + d^t, t' = a^x + f>il/ + c^z + d^t (5) gives us, provided its determinant is not zero, a one-to-one trans- formation of the points of space, which carries over planes into planes, and therefore also straight lines into straight lines, and is called a collineation or projective transformation of space. The same idea can be extended to spaces of higher dimensions. Quite as important is the case of one dimension. The transfor- mation (6) x' = a-^x + bjt, t' = a^x + b^t gives us, provided its determinant is not zero, a one-to-one trans- formation of the points on a line. This we call a projective trans- formation of the line, the term collineation being in this case obviously inadequate. It is possible, although for most purposes not desirable, to express the projective transformations (6), (1), (5) in one, two, and three dimensions in terms of non-homogeneous, instead of homogeneou? coordinates. We thus get the formulse (7) X' = a„X+ h' (8) j^i _ ayX+ h-^Y+c^ a^X + h^Y + c^ y,_ a^X-\-h^Y+e^ a^X+boY+c^ (9) ^1 __ a^X +b^Y+c^Z+ c?i a^X+b^Y+c^Z+d^ , _ a^X +b^Y+c^Z+ d^ a^X+b^Y+c^Z+dl a„X+ b„Y+ CoZ+d^ Z' = a^X+b^Y+c^Z+d^ These fractional forms may, in particular, be used to advantage in case their denominators reduce to mere constants. This special case, which is known as an affine transformation, may clearly be char- acterized by saying that all finite points go into finite points.* * If we consider the still more special case in which the constant terms in th* numerators of (8) and (9) are zero, that is, affine transformations in which the origin is transformed Into itself, we see that our formulae (8) and (9) h'^e the form (6) and LINEAR TRANSFORMATIONS AND MATRICES 71 These affine transformations are of much importance in mechanics, where they are known as homogeneous strains; cf., for instance, Webster's Dynamics (Leipzig, Teubner), pp. 427-4:^40. Although we propose to leave the detailed discussion of singular transformations to the reader (see Exercise 1 at the end of this sec- tion), we will give one theorem concerning them. Theorem 1. ^ the points Pj, Pj' '" ^'"^ carried over by a singu- lar projective transformation into the points Pj, Pg, •■•. then, if our transformation is in one dimension, the points P' will all coincide ; if in two dimensions, they will all he collinear ; if in three dimensions, they will all be complanar, etc. Suppose, for instance, that we have to deal with two dimensions. Since the determinant of the coUineation (1) is supposed to be zero, the three polynomials in the second members of (1) are linearly de- pendent ; that is, there exist three constants, Aj, k^, A3, not all zero, and such that for all values of x, y, t, (10) k^x' + k^y' + k/ = 0. Accordingly all points («', y', t') obtained by this transformation lie on the line (10). Similar proofs apply to the cases of one dimension and of three or more dimensions. Theorem 2. Any three distinct points on a line may be carried over respectively into any three distinct points on the line by one, and only one, projective transformation. Let the three initial points be P^, P^, P3, with homogeneous coor- dinates (a^i, «i), {x^, f^), (x^, t^) respectively, and let the points into which we wish them transformed be PJ, P^, P3 with coordinates {x[, t[), (4, 4), (4, ig'). The projective transformation x' = ax + ^t, t' = ^x+ U (1) respectively. Thus (6) may be regarded either as the general projective transfor- mation of a line (if ^, t are regarded as homogeneous coordinates) or as a special afane transformation of the plane (if x, t are regarded as non-homogeneous coordi- nates). Similarly (1) may be regarded either as the general projective transformation of a plane, or as a special afBrie transformation of space. 72 INTRODUCTION TO HIGHER ALGEBRA carries over any given point {x, t) into a point {x', t') whose position depends on the values of the constants a, 13, y, S. Our theorem is true if it is possible to find one, and, except for a constant factor which may be introduced throughout, only one, set of seven con- stants — four, a, y3, 7, S, and three others, pj, p^, p^, none of which is zero — which satisfy the six equations ^Pjx'^ = axi + /3«i, 1^24 = «a;2 + /3«2. fPs^'s = '^H + /3«3, \pA = 'iH + ^^1' [PiA = 72:2 + ^*r [Pa^'a = 7^3 + ^h- Since the aj's and t's are all known, we have here six homogeneous linear equations in seven unknowns. Hence there are always solu- tions other than zeros, the number of independent ones depending on the rank of the matrix of the coefficients. Transposing and rear- ranging the equations, we have Xja + ij/S -^iPl x^a + 1^ a^aV + *a^ aJjttH-^g/S 2:37+ <3^ = 0, = 0, - ^W = <^» - «^a = 0, - ^zPz = 0, -«^3 = 0. The matrix of these equations is of rank six. For consider the determinant of the first six columns with its sign reversed, 2) = Since Pj, Pj, Pg are distinct, there exist two constants Cj, c^ neither of which is zero, such that "I'^'i ' "2^2 H" 273 = U, ^1 h x\ H h X ^1 h A «2 h ^^ *3 h Xr, u LINEAR TRANSFORMATIONS AND MATRICES 73 Hence, adding to the fifth row of D Cj times the first row and c^ times the second, and to the sixth row c^ times the third row and c^ times the fourth, we have D = ^1 <1 a.1 H h 4 ^1 *1 = CiC2 2 44 ti4 oA oA cA cA and this is not zero, since P^ and P'^ are distinct as well as Pj and Pj. In the same way we see that the determinants obtained by striking out the sixth and the fifth columns respectively of the matrix are not zero. Accordingly, by Theorem 4, § 17, we see that the equations have a solution in which none of the quantities jOj, p^, /O3 are zero, and that every solution is proportional to this one. All these solutions obviously yield the same projective trans- formation of the line. CoBOLLARY. The transformation J ust determined is non-singular. This follows, by a reference to Theorem 1, from the fact that it does not carry Pj, Pg, P3 into a single point. EXERCISES 1. Discuss singular projective transformations in one, two, and three dimen- sions ; noting, in particular, the effect of the rank of the matrix of the transfor- mation, first, on the distribution of the points which have no corresponding points after the transformation, and secondly, on the distribution of the points into which no points are carried over by the transformation. 2. Prove that any four complanar points no three of which are collinear may be carried over into any four points in the plane, no three of which are collinear, by one and only one collineation. 3. State and prove the corresponding theorem in n dimensions. 4. Prove that the transformation from a first system of homogenecms coordi- nates to a second is effected by a non-singular linear transformation. Considei the case of one, two, and three dimensions. 74 INTRODUCTION TO HIGHER ALGEBRA 5. Prove that a projective transformation in space effects on every plane a two-dimensional, and on every line a one-dimensional, projective transformation, while at the^same time the positions of the plane and line are changed. [SnoGESTioN. If p and pi are any two corresponding planes, assume in any way a pair of perpendicular axes in each of them, and denote hy (a;^, y^, t^, and (xL y' t') respectively the systems of two-dimensional homogeneous coordinates based on these axes. Then show, by using the result of Exercise 4, that the transformation of one plane on the other will be expressed by writing x[, y!^, t^ as homogeneous linear poly- nomials in Xj, y^, «j.] 25. Further Development of the Algebra of Matrices. We proceed to establish certain further properties of matrices, leaving, however, much to the reader in the shape of exercises at the end of the section. The theory of linear transformations suggests to us at once certain properties of matrices. The first of these is : Theorem 1. The matrix 1 1 = has the property that if a is any matrix whatever ■ la = al = a. For the linear transformation of which a is the matrix will evi- dently not be changed by being either followed or preceded by the identical transformation of which I is the matrix. If we do not wish to use the idea of linear transformation, we may prove the theorem directly by actually forming the products la and al. This theorem tells us that I plays in the algebra of matrices the same role that is played, by 1 in ordinary algebra. For this reason I is sometimes called the unit matrix or idemfactor. Let us now consider any non-singular linear transformation and its inverse. These two transformations performed in succession in either order obviously lead to the identical transformation. This gives us the theorem : Theorem 2. If a = ni' *nl *!» LINEAR TRANSFORMATIONS AND MATRICES T5 18 a non-singular matrix of determinant a, and if A^j- denote in the ordi nary way the cof actors of the elements of a, the matrix a a a a called the inverse of a, and denoted hy a"S is a non-singular matrix which has the property that aa~i = a~ia = I. This suggests that we define positive and negative integral powers of matrices as follows : Definition 1. If p is any positive integer and a any matrix we understand hy aP the product aa • • • a to p factors. If a. is a non- singular matrix, we define its negative and zero powers hy the for- mulce a.-p = (a-iy, a" = I. From this definition we infer at once Thboeem 3. 77ie laws of indices aPa." = a^+«, {aPf = aP^ hold for all matrices when the indices p and q are positive integers, and for all non-singular matrices when p and q are any integers. We turn now to the question of the division of one matrix by another. We naturally define division as the inverse of multiplica- tion, and, since multiplication is not commutative, we thus get two distinct kinds of division ; a divided by b being on the one hand a matrix x such that „ . a = Dx, on the other hand a' matrix y such that a = yb. On account of this ambiguity, the term division is not ordinarily used here. We have, however, as is easily seen, the following theorem : 76 INTRODUCTION TO HIGHER ALGEBRA Theorem 4. If & is any matrix and b any non-singular matrix^ there exists one, and only one, matrix x which satisfies the equation a = bx, and one, and only one, matrix y which satisfies the equation a = yb, and these matrices are given respectively hy theformulce. X = b~'a, y = ab~^ A special class of matrices is of some importance ; namely, those of the type h . • h ■ • • • • • k Such matrices we will call scalar matrices for a reason which will presently appear. If we denote by k the scalar matrix just written, and by a aijv matrix of the same order as k, we obtain readily the formula (1) ka = ak = ka.. If now, besides the scalar matrix k, we have a second scalar matrix 1 in which each element in the principal diagonal is I, we have the two formulae (2) k + l = H-k = (^ + OI, (3) kl = lk = Hl. Formula (1) shows that scalar matrices may be replaced by ordinary scalars when they are to be multiplied by other matrices ; while formulae (2) and (8) show that scalar matrices combined with one another not only obey all the laws of ordinary scalars, but that each scalar matrix may in such cases be replaced by the scalar which occurs in each element of its principal diagonal provided that at the end of the work the resulting scalar be replaced by the correspond- ing scalar matrix LINEAR TRANSFORMATIONS AND MATRICES 77 For these reasons we may, in the algebra of matrices, replace all scalar matrices by the corresponding scalars, and then consider that all scalars which enter into our work stand for the corresponding scalar matrices. If we do this, the unit matrix I will be represented by the symbol 1. Definition 2. By the adjoint A. of a matrix a is understood another matrix of the same order in which the element in the ith row and jth column is the cofaetor of the element in the jth row and ith column of a.* It will be seen that when a is non-singular, (4) A = aa-i, but it should be noticed that while every matrix has an adjoint, only non-singular matrices have inverses. Equation (4) may be written in the form j5) Aa = aA = aI, a form in which it is true not merely when a is non-singular, but also, as is seen by direct multiplication, when the determinant of a is zero. Finally we come to a few important theorems concerning the rank of the matrix obtained by multiplying together two given matrices. In the first place, we notice that the rank of the product is not always completely determined by the ranks of the factors. This may be shown by numerous examples, for instance, in formula (5), § 22, the ranks of the factors are in general two and one, and the rank of the product is zero, while in the formula ^11 hi ^31 *12 •*32 , 1 = *12 *32 the ranks of the factors are in general the same, namely two and one, while the rank of the product is one. But though, as this example shows, the ranks of the factors (even together with the order of the matrices) do not suffice to determine the rank of the product, there are, nevertheless, important inequali- ties between these ranks, one o^ which we now proceed to deduce. * Notice the Interchange of rows and columns here, which in the case of adjoint determinants, being immaterial and sometimes inconvenient, was not made. 78 INTRODUCTION TO HIGHER ALGEBRA For this purpose consider the two matrices a = "u *i» b = *n •■• K "m and their product ab. Theorem 5. Any k-rowed determinant of the matrix ab is equal to an aggregate of k rowed determinants of b each multiplied into a polynomial in the as, and also to an aggregate of k-rowed determinants of a each multiplied hy a polynomial in the b's. For any A-rovved determinant of ab niay be broken up into a sum of determinants of the A:th order in such a way that each column of each determinant has one of the b's as a common factor.* After taking out these common factors from each determinant, we have left a determinant in the a's which, if it does not vanish identically, is a ^-rowed determinant of a. Or, on the other hand, we may break up the A;-rowed determinant of ab into a sum of determinants of the kth order in such a way that each row of each determinant has one of the as as a common factor. After taking out these common factors from each determinant, we have left a determinant in the b's which, if it does not vanish identically, is a A-rowed determinant of b. From the theorem just proved it is clear that if all the A-rowed determinants of a or of b are zero, the same will be true of. all the A-rowed determinants of ab. Hence Theorem 6. TJie rank of the product of two matrices cannot exceed the rank of either factor, f * The truth of this statement and the following will be evident if the reader actually writes out the matrix ab. t Thus if j'l and r« are the ranks of the two factors and B is the rank of the prod- uct, we have J?<ri, i?<r2. This is one half of Sylvester's "Law of Nullity," of which the other half may be stated in the form It^r\ + r^ — n, where n is the order of the matrices ; cf. Exercise 8 at the end of this section. Sylvester defines the nullity of a matrix as the difference between its order and its rank, so that his statement of the law of nullity is : The nullity of the product of two matrices is at least as gi-eat as the nullity of either factor, and at most as great as the sum of the nullities of t)>« factors. LINEAR TRANSFORMATIONS AND MATRICES 79 There is one important case in which this theorem enables us to determine completely the rank of the product, namely, the case in which one of the two matrices a or b is non-singular. Suppose first that a is non-singular, and denote the ranks of b and ab by r and i2 respectively. By Theorem 6, R<r. We may, however, also regard b as the product of a~^ and ab, and hence, applying Theorem 6 again, we have r^R. Combining these two results, we see that r = B. On the other hand, if b is non-singular, and we denote the ranks of a and ab respectively by r and R, we get from Theorem &,R<r; and, applying this theorem again to the equation (ab) b-i = a, we have r<R. Thus again we get r = R. We have thus established the result : Theorem 7. If a matrix of rank r is multiplied in either order hy a non-singular matrix, the rank of the product is also r. EXERCISES 1. Prove that a necessary and sufficient condition that two matrices a and b of the same order be equivalent is that there exist two non-singular matrices c and d such that , . dac = D. Cf. § 22, Exercise 2, and § 19, Exercise 4. 2. Prove that a necessary and sufficient condition that two matrices a and b of the same order be equivalent is that there exist four matrices c, d, e, f such ^"^^^ dac = b, a = fbe. 3. Prove that every matrix of rank r can be written as the sum of r matrices of rank one.* [Suggestion. Notice that the special matrix mentioned in § 19, Exercise 3, can be so written.] * A matrix of rank one has been called by Gibbs a dyad, since it may (cf. § 19, Ex. 5) be regarded as a product of two complex quantities (ai, a^, ■■• fln) and (6i, 62, ••• M- The sum of any number of dyads is called a dyadic polynomial, or simply a dyadic. Every matrix is therefore a dyadic, and vice versa. Gibbs's theory of dyadics, in the case ra = 3, is explained in the Vector Analysis of Gibbs-Wilson, Chap. "V. Geometric language is used here exclusively, the complex quantities (Oi, (i2, as) and (61, 62, 63) from which the dyads are built up being interpreted as vectors in space of three dimensions. This theory is equivalent to Hamilton's theory of the Linear Vector Function in Quaternions. 80 INTRODUCTION TO HIGHER ALGEBRA 4. Prove that a necessary and sufficient condition that a matrix be a divisor of zero (cf . § 22, Exercise 1) is that it be singular. [SnoGESTiON. Consider equivalent matrices.] 5. Prove that the inverse of the product of any number of non-singular matrices is the product of the inverses of these matrices taken in the reverse order. Hence deduce a similar theorem concerning the adjoint of a product of any number of matrices, whether these matrices are singular or not. What theorem concerning determinants can be inferred ? 6. Prove that the conjugate of the inverse of a non-singular matrix is the inverse of the conjugate ; and that the conjugate of the adjoint of any matrix is the adjoint of the conjugate. 7. Prove that if a matrix has the property that its product vrith every matrix of the same order is commutative, it is necessarily a scalar matrix. 8. If ri and r^ are the ranks of two matrices of order n, and R the rank of their product, prove that i?-> i * [Suggestion. Prove this theorem first on the supposition that one of the two matrices which are multiplied together is of the form mentioned in Exercise 3, § 19, using also at this point Exercise 1, § 8. Then reduce the general case to this one by means of Exercise 1 of this section.] 26. Sets, Systems, and Groups. These three words are the technical names for conceptions which are to be met with in all branches of mathematics. In fact the first two are of such gener- ality that they may be said to form the logical foundation on which all mathematics rests, f In this section we propose, after having given a brief explanation of these three conceptions, to show how they apply to the special subjects considered in this chapter. The objects considered in mathematics — we use the word ohject in the broadest possible sense — are of the most varied kinds. We have, on the one hand, to mention a few of the more important ones, the different kinds of quantities ranging all the way from the posi- tive integers to complex quantities and matrices. Next we have in geometry not only points, lines, curves, and surfaces but also such * Cf. the footnote to Theorem 6. t For a popular exposition of the point of view here alluded to, see my address on The Fundamental Conceptions and Methods of Mathematics, St. Louis Congress oi Arts and Science, 1904. Reprinted in Bull. Amer. Math. Soc. , December, 1904. LINEAR TRANSFORMATIONS AND MATRICES 81 things as displacements (rotations, translations, etc.), collineations, and, in fact, geometrical transformations in general. Then in vari- ous parts of mathematics we have to deal with the Theory of Substitutions, that is, with the various changes which can be made in the order of certain objects, and these substitutions themselves may be regarded as objects of mathematical study. Finally, in mechanics we have to deal with such objects as forces, couples, velocities, etc. These objects, and all others which are capable of mathematical consideration, are constantly presenting themselves to us, not singly, but in sets. Such sets (or, as they are sometimes called, classes) of objects may consist of a "finite or an infinite number of objects, or elements. We mention as examples : (1) All prime numbers. (2) All lines which meet two given lines in space. (3) All planes of symmetry of a given cube. (4) All substitutions which can be performed on five letters. (5) All rotations of a plane about a given line perpendicular to it. Having thus gained a slight idea of the generality of the con- ception of a set, we next notice that in many cases in which we have to deal with a set in mathematics, there are one or more rules by which pairs of elements of the set may be combined so as to give objects, either belonging to the set or not as the case may be. As examples of such rules of combination, we mention addition and multiplication both in ordinary algebra and in the algebra of ma- trices ; the process by which two points, in geometry^ determine a line ; the process of combining two displacements to give another displacement, etc. Such a set, with its associated rules of combination, we will call a mathematical system, or simply a system.* We come now to a very important kind of system known as a group, which we define as follows : * This definition is sufficiently general for our immediate purposes. In general, however, it is desirable to admit, not merely rules of combination, but also relations be- tween the elements of a system. In fact we may have merely one or more relations and no rules of combination at all. From this point of view the positive integers with the relation of greater and less would form a system, even though we do not introduce any rule of combination such as addition or multiplication. It may be added that rules of combination may be regarded as merely relations between three objects ; of. the address referred to above. 82 INTRODUCTION TO HIGHER ALGEBRA Definition. A si/stem consisting of a set of elements and one ruU of combination, which we will denote hy o, is called a group if thefolloiv- ing conditions are satisfied : (1) If a and b are any elements of the set, whether distinct or not, aob is also an element of the set.* (2) The associative law holds ; that is, if a, b, c are any elements of the set, {aob)oc = ao{boc). (3) The set contains an element, i, called the identical element, which is such that every element is unchanged when combined with it, ioa=aoi = a. (4) Jf a is any element, the set also contains an element a', called the inverse of a, such that a'oa = aoa' = i. Thus, for example, the positive and negative integers with zero form a group if the rule of combination is addition. In this case zero is the identical element, and the inverse of any element is its negative. These same elements, however, do not form a group if the rule of combination is multiplication, for while conditions (1), (2), and (3) are fulfilled (the identical element being 1 in this ease), condition (4) is not, since zero has no reciprocal. Again, the set of all real numbers forms a group if the rule of combination is addition, but not if it is multiplication, since in this case zero has no inverse. If we exclude zero from the set, we have a group if the rule of combination is multiplication, but not if it is addition. As an example of a group with a finite number of elements we mention the four numbers + 1, - 1, + V^ - V^Tl with multiplication as the law of combination. In order to get an example of a group of geometrical operations, let us consider the translations of a plane, regarded as a rigid lamina, in the directions of its own lines. Every such translation may be represented both in magnitude and in direction by the length and * A system satisfying condition (1) is sometimes said to have "the group prop- erty." In the older worlts on the subject this condition was the only one to be explicitly mentioned, the others, however, being tacitly assumed. LINEAR TRANSFORMATIONS AND MATRICES 83 direction of an arrow lying in the plane in question. Two such translations performed in succession are obviously equivalent to a translation of the same sort represented by the arrow obtained by combining the two given arrows according to the law of the paral- lelogram of forces. The set of all translations with the law of combination just explained is readily seen to form a group if we include in it the null translation, i.e. the transformation which leaves every point in the plane fixed. This null translation is then the identical element, and two translations are the inverse of each other if they are equal in magnitude and opposite in direction. All the groups we have so far mentioned satisfy, not only the four conditions stated in the definition, but also a fifth condition, viz. that the law of combination is commutative. Such groups are called commutative or Ahelian groups. In general, however, groups do not have this property. As examples of non-Abelian groups, we may mention first the group of all non-singular matrices of a given order, the rule of combination being multiplication ; and secondly the group of all matrices of a given order whose deter- minants have the value ± 1, the rule of combination being again multiplication. This second group is called a subgroup of the first, since all its elements are also elements of the first group, and the rule of combination is the same in both cases. A subgroup of the group last mentioned is the group of all matrices of a given order whose determinants have the value -1~ 1,* the rule of combination being multiplication. We add that non-Abelian groups may readily be built up whose elements are linear transformations, or coUineations. On, the other band, Abelian groups may be formed from matrices if we take as our ' ule of combination addition instead of multiplication, 27. Isomorphism. Definition. Two groups are said to be isomorphic^ if it is possible to establish a one-to-one correspondence between their elements of such a * These are called unimodular matrices ; or, more accurately, properly unimodular matrices to distinguish them from the improperly unimodular matrices whose determi- nants have the value — 1. It should be noticed that these last matrices taken by them- selves do not constitute a group, since they do not even have the group property. t Simply isomorphic would be the more complete term. We shall, however, not be concerned with isomorphism which is not simple. 84 INTRODUCTION TO HIGHER ALGEBRA gort that if a, h are any elements of the first group and a\ h' the corre sponding elements of the second, then a' o b' corresponds to aob.* We proceed to illustrate this definition by some 'examples, leaving to the reader the proofs of the statements we make. In each case we omit the statement of the rule of combination in the case of transformations, where no misunderstanding is possible. FiEST Example, (a) The group of the four elements 1, V-1, -1, -V-1, the rule of combination being multiplication. (b) The group of four rotations about a given line through angles of 0°, 90°, 180°, 270°. ' These two groups may be proved to be isomorphic by pairing the elements against one another in the order in which they have just been written. Second Example, (a) The group of the four matrices (o i),(~o -i),(o -l),( l), the rule of combination being multiplication. (6) The group of the following four transformations : the iden- tical transformation ; reflection in a plane ; reflection in a second plane at right angles to the first ; rotation through 180° about the line of intersection of these two planes. (c) The group consisting of the identical transformation and of three rotations through angles of 180° about three straight lines through a point at right angles to one another. The two groups of Example 1 are not isomorphic with the three of Example 2 in spite of the fact that there are the same number of ele- ments in all the groups. This follows from the presence of two elements in the groups of Example 1 whose squares are not the identical element. * This idea of isomorphism may obviously be extended to the case of any two sys- tems provided merely that there are the same number ofrules of combination in both cases. Thus the system of all scalar matrices on the one hand and of all scalars on the other, the rules of combination being in both cases addition and multiplication, are ob- • viously isomorphic. It is for this reason that no confusion arises if no distinction la made between scalar matrices and scalars. LINEAR TRANSFORMATIONS AND MATRICES 85 Third Example, (a) The group of all real quantities ; the rule of combination being addition. (J) The group of all scalar matrices of order h with real ele- ments ; the rule of combination being addition. (e) The group of all translations of space parallel to a given line. Fourth Example, (a) The group of all non-singular matrices of order n, with multiplication as the rule of combination. (5) The group of all non-singular homogeneous linear transfor- mations in n variables. We might be tempted to mention as a group of geometrical trans- formations isomorphic with the last two groups, the group of all non-singular coUineations in space of w — 1 dimensions. This, how- ever, would be incorrect, for the correspondence we have established between coUineations and linear transformations is not one-to-one ; to every linear transformation corresponds one coUineation, but to every coUineation correspond an infinite number of linear transfor,- mations, whose coefficients are proportional to one another.* In order to get a group of geometrical transformations isomorphic with the group of non-singular matrices of the nth order it is sufficient to interpret the variables x-^, ■■■ x„ as non-homogeneous coordinates in space of n dimensions, and to consider the geometric transformation effected by non-singular homogeneous linear transformations of these x's. These transformations are those affine transformations of space of n dimensions which leave the origin unchanged ; cf. the footnote on p. 70. Thus the group of all non-singular matrices of the nth order is isomorphic with a certain subgroup of the group of coUineations in space of n dimensions, not with the group of all non-singular coUineations in space of w — 1 dimensions. An essential difference between these two groups is that one * This does not really prove that the groups are not isomorphic, since it is con- ceivable that some other correspondence might be established between their elements ■which would be one-to-one and of such a sort as to prove isomorphism. Even the fact, to be pointed out presently, that the groups depend on a different number of parameters does not settle the question. A reference to the result stated in Exercise 7, § 2.'>, shows that the groups are not isomorphic ; for, according to it, the only non- singular coUineation which is commutative with all coUineations is the identical transformation, whereas all linear transformations with scalar matrices havp this property. mTRODUCTION TO HIGHER ALGEBRA depends on y? parameters (the i^ coefficients of the linear transforma- tion' while the other depends only on i^ — 1 parameters (the ratios of the coefficients of the coUineation). We can, however, by looking at the subject a little differently, obtain a group of matrices isomorphic with the group of all non- singular coUineations in space of m — 1 dimensions. For this purpose we need merely to regard two matrices as equal whenever the ele- ments of one can be obtained from those of the other by multiplying all the elements by the same quantity not zero. When we take this point of view with regard to matrices, it is desirable to indicate it bj"- a new terminology and notation. According to a suggestion of E. H. Moore of Chicago, we will call such matrices fractional matrices, and write them , etc. Agreeing that fractional matrices are. to be added and mul- tiplied according to the same rules as ordinary matrices, we may now say that the group of all non-singular coUineations in space of w — 1 dimensions is isomorphic with the group of all fractional matrices of the wth order whose determinants are not zero.* To take another example, the groups in the second example above are isomorphic with the group whose elements are the four fractional matrices «11 «12 «11 «12 «13 1 «31 «22 «23 «21 «22 «31 «32 «33 1 1 -1 1 1 1 1 and where the law of combination is multiplication. These four matrices, if regarded as ordinary matrices, would not even satisfy the first condition for a group. The reader wishing to get a further insight into the theory of groups of linear transformations will find the following three treat- * It should be noticed that we cannot speak of the value of the determinant of a fractional matrix unless this value is zero, for if we multiply all the elements of the matrix by c we do not change the matrix, but do multiply the determinant by c": There is in particular no such thing as a unimodular fractional matrix. We may however, speak of the rank of a fractional matrix. LINEAR TRAKSiORMATlONS AND MA'I'RICES 8? ments interesting and instructive. They duplicate each other to only a very slight extent. Weber, Algebra, Vol. II. Klein, Vbrlesungen uher das Ihosaeder. Lie-Scheffers, Vorlesungen iiber continuirliche Grv/ppen. EXERCISES 1. Definition. A group is said to be of order n if it contains n, and only n, elements. If a group of order n has a subgroup, prove that the order of this subgroup is a factor of n. [Suggestion. Denote the elements of the subgroup by ai ••• at, and let 6 be any other element of the group. Show that bau baz, ■■■ bat are all elements of the group distinct from each other and distinct from the a's. If there are still other elements, let c be one and consider the elements cai, ■■■ ea^, etc.] 2. Prove that if a is any element of a group of finite order, it is possible by multiplying a by itself a sufficient number of times to get the identical element. Definition. The lowest power to which a can be raised so as to give the identical element is called the period of a. 3. Prove that every element of a group of order n has as its period a factor of n (I and n included). 4. Definition. A group is called cyclic if all its elements are powers of a single element. Prove that all cyclic groups of order n are isomorphic with the group of rota- tions about an axis through angles 0, w, 2 <o, — (n — 1) to, vrhere to = 2 ■n-/n, and that conversely every such group of rotations is a cyclic group. 5. Prove that every group whose order is a prime number is, a cyclic group. 6. Prove that all groups of order 4 are either cyclic or isomorphic with the groups of the second example above. A group of this last kind is called a fours group (^Vierergruppe"). 7. Obtain groups with regard to one or the other of which all groups of order 6 are isomorphic. 8. Obtain groups with regard to one or the other of which all groups of order 8 are isomorphic. CHAPTER VII INVARIANTS. FIRST PRINCIPLES AND ILLUSTRATIONS 28. Absolute Invariants, Geometric, Algebraic, and Arithmetical. If we subject a geometric figure to a transformation, we find that, while many properties of the figure have been altered, others have not. If we consider, not a single transformation, but a set of trans- formations, then those properties of figures which are not changed by any of the transformations of the set are said to be invariant prop- erties with regard to this set of transformations. Thus if our set of transformations is the group of all displacements, the property of two lines being parallel or perpendicular to each other and the property of a curve being a circle are invariant properties, since after the transformation the lines will still be parallel or perpendicu- lar and the curve will still be a circle. If, however, we consider, not the group of displacements, but the group of all non-singular coUineations, none of the properties just mentioned will be invariant properties. Properties invariant with regard to all non-singular coUineations have played such an important part in the development of geometry that a special name has been given to them, and they are called projective or descriptive properties. As examples of such projective properties we mention the collinearity and complanarity of points, the complanarity and concurrence of lines, etc.; or, on the other hand, the contact of a line with a curve or a surface or the contact of two curves or of two surfaces, or of a curve and a surface. Definition 1. If there is associated with a geometric figure a quantity which is unchanged hy all the transformations of a certain set, then this quantity is called an invariant with regard to the trans- formations of the set. For instance, if our set of transformations is the group of dis- placements, the distance between two points and the angle between two lines would be two examples of invariants. 88 INVARIANTS. FIRST PRINCIPLES AND ILLUSTRATION* 89 The geometric invariants so far considered lead up. naturally to the subject of algebraic invariants. Thus let us consider the two polynomials U,. + £,,+ C,, [A^x + B^y + G^, and subject the variable^ (a;, y) to the transformations of the set (2) x' = X cos6 + y smO + a, y' = — xsin + y cos 6 + ^, where «, /3, are parameters which may have any values. The trans- formation (2) carries over the polynomials (1) into two new poly- nomials : .3) lA',x' + B[y'+C[, \A'^x' + £',y'+Oi. The coefficients of (3) may be readily expressed in terms of the coeffi- cients of (1) and the parameters a, /8, 6. Using these expressions, we easily obtain the formulas .4. .UiB>,-A',Bi = A,B,-A,B„ [A[Al + B[Bi = A^A, + B,B^. We shall therefore speak of the two expressions (5) A,B^-A^B,, A,A^+B^B^ as invariants of the system of polynomials (1) with regard to the set of transformations (2) according to the following general definition : Definition 2. If we have a system of polynomials in the variables {x, y, z, ...) arid a set of transformations of these variables, then any function of the coefficients of the polynomials is called an invariant (or more accurately an absolute invariant^ with regard to these transforma- tions if it is unchanged when the polynomials are subjected to all the transformations of the set. The relation of the example considered above to the subject of geometric invariants "becomes obvious when we notice that the alge- braic transformations (2) may be regarded as expressing the dis- placements of plane figures in their plane when (x,y) are rectangular coordinates of points in the plane. If now we consider, not the poly- nomials (1), but the two lines determined by setting them equal tc 90 . INTRODUCTION TO HIGHER ALGEBRA zero, we have to deal with the displacements of these two lines. The invariants (5) have themselves no geometric significance, but by equating them to zero, we get the necessary and sufficient conditions that the two lines be respectively parallel and perpendicular, and these, as we noticed above, are invariant properties with regard to displacements. Finally we may notice that the ratio of the two in- variants (5) gives the tangent of the anglfe between the lines, — a geometric invariant. As a second example, let us consider, not two lines, but a line and a point. Algebraically this means that we start with the system ... Ux+By+O, consisting of a polynomial and a pair of variables. We shall wish to demand here that whenever the variables (x, y) are subjected to a transformation, the variables (x^, y^ be subjected to the same trans- formation, or as we say according to Definition 3 below, that {x, y) and (aTj, y-^ be cogredient variables. If we subject the system (6) to any transformation of the set (2), we get a new system ,„. {A'a/ + B'y'+C', and it is readily seen that A'x[ + B'yi +C' = Ax^ + By^ + O. Accordingly we shall call Ax.^^ + By-^ + C a covariant of the system (6) according to Definition 4 below. This covariant has also no direct geometric meaning, but its vanishing gives the necessary and suffi- cient condition for an invariant property, namely, that the point {xy, yi) lie on the line Ax + By +0=0. In the light of this example we may lay down the following gen- eral definitions : Definition 3. If we have several sets of variables (x, y, 2, ■■■), («i, «/i, 2i, •••), X^2' 1/v «'2' •••)' — and agree that whenever one of these sets is subjected to a transfor- mation every other set shall be subjected to the same transformation, then we say that we have sets of cogredient variables. INVARIANTS. FIRST PRINCIPLES AND ILLUSTRATIONS 91 Definition 4. If we have a system consisting of a number of poly- nomials in (x, y, z, ■••) and of a number of sets of variables co- gredient to (x, y, s, •■•), then any function of the coefficients of the polynomials and of the cogredient variables which is unchanged when the variables (x, y, z, ••■) are subjected to all the transformations of a certain set is called a covariant {or more accurately an absolute cova- riant) of this system with regard to the transformations of this set. It will be seen that invariants may be regarded as special cases of covariants. Among the geometric invariants there are some which from their nature are necessarily integers, and which we will speak of as arith- metical invariants. An example would be the number of vertices of a polygon if our set of transformations was either the group of dis- placements or the group of non-singular coUineations. Another ex- ample is the largest number of real points in which an algebraic curve can be cut by a line, if our set of transformations is the group of reac non-singular coUineations. These arithmetical invariants also play, as we shall see, an impor- tant part in algebra. We mention here as an example the degree of an w-ary form, which is an invariant with regard to all non-singular linear transformations. * EXERCISES 1. Prove that (x^ - xiy + (y^ - ViY, and are covariants of the system (xi, yi), (12, y^), (xs, ys) with regard to the transformations (2). 2. Prove that A+B and B^-AC are invariants of the polynomial Ax^ + 2 Bxy + Cy^ + 2 Dx +2 Ey + F with regard to the transformations (2). What geometric meaning can be attached to these invariants? 3. Prove that A^ + B'' is an invariant of the polynomial Ax + By+C with regard to the transformations (2). Hence show that Axi + Byi + C VA^ + B^ is a covariant of the system (6). Note its geometric meaning. *It is, in fact, an invariant with regard to all linear transformations except the one in which all the coefficients of the transformation are zero. X\ n 1 X2 y^ 1 Xi y 1 92 INTRODUCTION TO HIGHER ALGEBRA 29. Equivalence. Definition 1. If A and B denote two geometric configuratiom or two algebraic expressions, or sets of expressions, then A and B shall he said to he equivalent with regard to a certain set of transformations when, and only when, there exists a transformation of the set which car- ries over A into B and also a transformation of the set which carries over B into A. To illustrate this definition we notice that the conception of equiv- alence of geometric figures with regard to displacements is identical with the Euclidean conception of the equality or congruence of fig- ures. Again, we see from Theorem 2, § 24, that on a straight line two sets of three points each are always equivalent with regard to non- singular projective transformations. In both of the illustrations just mentioned the set of transforma- tions forms a group. In such cases the condition for equivalence can be decidedly simplified, for the transformation which carries A into B has an inverse belonging to the set, and this inverse necessarily carries B into A. Thus we have the Theokem. a necessary and sufficient condition for the equiva- lence of A and B with regard to a group of transformations is thatji transformation of the group carry over A into B. This theorem will be of great importance, as the question of equivalence will present itself to us only when the set of transfor- mations we are considering forms a group. Let us consider, for the sake of greater definiteness, a group of geometric transformations. If two geometric configurations are equivalent with regard to this group, every invariant of the first configuration must be equal to the corresponding invariant of the second. Thus, for instance, if two triangles are equivalent with re- gard to the group of displacements, all the sides and angles of the first will be equal to the corresponding sides and angles of the second. The same will be true of the altitudes, lengths of the medial lines, radius of the inscribed circle, etc., all of these being invariants. Now one of the first problems in geometry is to pick out from among these invariants of the triangle as small a number as possible whose equality for two triangles insures the equivalence of the triangles. This can be done, for instance, by taking two sides and the included INVARIANTS. FIRST PRINCIPLES AND ILLUSTRATIONS 93 angle, or two angles and the included side, or three sides. Any one of these three elements may be called a complete system of invariants for a triangle with regard to the group of displacements, since two triangles having these invariants . in common are equivalent and therefore have all other invariants in common. The conception we have here illustrated may be defined in general terms as follows : Definition 2. A set of invariants of a geometric configuration or an algebraic expression are said to form a complete system of invariants if two configurations or expressions having these invariants in common are necessarily equivalent.* It will be seen from this definition that all the invariants of a geometric configuration or of an algebraic expression are uniquely determined by any complete system of invariants. Finally we will glance at an application to matrices of the ideas of invariants and equivalence. Let us consider matrices of the nth order,! and consider transformations of the following form which transform the matrix A into the matrix B: (1) aAb = B, where a and b are any non-singular matrices of the nth. order. This transformation may be denoted by the symbol (a, b), and these sym- bols must obviously be combined by the formula » By means of this formula it may readily be shown that these trans- formations form a group. According to our general definition of equivalence, two matrices A and B must therefore be said to be equivalent when, and only when, two non-singular matrices a and b exist which satisfy (1). That this definition of equivalence amounts to the same thing as our earlier definition is seen by a reference to Exercise 1, § 25. * In the classical theory of algebraic invariants this term is used in a different and much more restricted sense. There we have to deal with integral rational relative inva- riants (cf . § 31) . By a complete system of such invariants of a system of algebraic forms is tiiere understood a set of such invariants in terms of which every invariant of this sort of the system of forms can be expressed integrally and rationally. Cf. for instance Clebseh, Binare Formen, p. 109. t We may, if we choose, confine our attention throughout to matrices with real elements. 94 INTRODUCTION TO HIGHER ALGEBRA 30. The Rank of a System of Points or a System of Linear Forms as an Invariant. Let (x^, y^. Zy, ^j), {x^, y^, Sj, t^, {x^., y^. Zg, t^ be any three distinct collinear points, so that the rank of the matrix ^1 VV % h is two. Now subject space to a non-singular QoUineation and we get three new points which will also be distinct and collinear, and hence the rank of their matrix will also be two. Thus we see that in this special case the rank of the system of points is unchanged by a non-singular coUineation. Again, let a^x + b^y + c^z + d^t = 0, a^x + b^y + e^z + d^t = 0, a^x + b^y + c^z + d^t = 0, a^x + b^+ c^z +d^t=0 be any four planes which have one, and only one, point in common, so that the rank of their matrix is three. After a non-singular coUin- eation we have four new planes which will also have one, and only one, point in common, and hence the rank of the matrix of their coefficients will be three. The rank of this system of planes is therefore unchanged by such a transformation. We proceed now to generalize these facts. Theorem 1. The rank of the matrix of m points {xf,xf, ...a;W), (;=1, 2, ... m) is an invariant with regard to non-singular linear transformations. Let [■ X^ = c^^Xj^ + • • • -f c^„x„, (1) -^M — ^nl^l + • ■ ■ + <^mv^n be a non-singular linear transformation which carries the points (xf, ••■ 2:5*1) over into the points {X^K ■■■ -X'M). Now suppose any k of the points (xf, ■■■ 4']), which for convenience we will take as the first k, are linearly dependent. Then there exist k constants (cj, ••• c^) not all zero, such that (2) c,x'^- + c^x'! +... + c^f = 0, 0' = 1, 2, .. . n). INVARIANTS. FIRST PRINCIPLES AND ILLUSTRATIONS 95 By means of the transformation (1) we have hence c^X'^ + c^X'J +■■■ + c^f = Cj^ (c^x'^ + c^{ + --+ c^^^) + ■■■ + Oj„(cix'^ + c^x':+-- +e^»i) (J = l, 2, -n). Since this vanishes on account of (2), the first k of the points {Xf, ■■■ X^'^) are linearly dependent. Since (1) is a non-singular transformation, it is immaterial which set of points we consider as the initial set. Thus we have shown that if any k points of either set are linearly dependent, the corresponding k points of the other set will be, also. Now if the rank of the matrix of the a;'s is r, at least one set of r of the 2;-points is linearly independent, but every set of (r+1) of them is linearly dependent. Consequently the same is true for the X-points, and therefore their matrix must also be of rank r. Theorem 2. The rank of the matrix of m linear forms /i(a;i, ■■■x,^ = a.i^i + ai^^-\ h ai^x,, (i = 1, 2, • • • wi) is an invariant with regard to non-singular linear transformations. The proof of this theorem, which is very similar to the proof of Theorem 1, we leave to the reader. It will be noticed that the invariants we have considered in this section are examples of what we have called arithmetical invariants. 31. Relative Invariants and Covariants. We will begin by con- sidering a system of n linear forms in n variables ^21 ■*'l "^ '''22 "^2 "•" ■ ■ ■ "1 '*2» •"»' (1) a«r 2^1 + «7!2 ^2 H 1" '^nn ^n- Definition 1. The determinant *ii *nl *i» is called the fesultant of the system (1). 96 INTRODUCTION TO HIGHER ALGEBRA Let us now subject the system (1) to the linear transformation ■ a^i = '^ii a;i + ■■• + ci„4, f2) This gives the new system of forms (3) where Cij- ' "T" ^in "ray From these formulae and the law of multiplication of matrices we infer that f4) "u n™ *ii *ln -11 -l" •*nl ■ ' ■ '^nn "nl ' " ' "'nn '^nl This result we state as follows : Theorem \. If a system of n linear forms in n variables with matrix a is subjected to a linear transformation with matrix c, the re- sulting system has as its matrix ac. Taking the determinants of both sides of (4), we see that the re- sultant of (1) is not an absolute invariant. It is, however, changed in only a very simple manner by a linear transformation, namely, by being multiplied by the determinant of the transformation. This leads us to the following definition : Definition 2. A rational function * of the coefficients of a form or system of forms which, when these forms are subjected to any non- singular linear transformation, is merely multiplied by the fith power (/u an integer f ) of the determinant of the transformation is called a rela- tive invariant of weight ft, of the form or system of forms. :f The forms themselves are called the ground forms. * Besides these rational invariants we may also consider irrational ones (cf. § 90), in which case the exponent /i will not necessarily be an integer. t The condition that n be an integer need not be included as a part of our hypothesis, since it may "be proved. The proof that ix cannot be a fraction is simple. The proof that /x cannot be irrational or imaginary would take us outside of the domain of algebra. t From this definition it is clear that every relative invariant is an absolute invariant with regard to the group of linear transformations of determinant +1. Cf , Exercise 7 . § 81. INVARIANTS. FIRST PRINCIPLES AND ILLUSTRATIONS 97 It will be seen that absolute invariants are simply relative in- variants of weight zero. The fact pointed out above concerning the resultant may now be stated in the following form : Theorem 2. The resultant of a set of n linear forms in n varidblet is a relative invariant of weight 1. We pass on now to relative covariants : Definition 3. If we have a system consisting of a number of n-ary forms and of a number of points {y^, ... «/„), (2j, ... 3„), ... , the coor- dinates of each of which are cogredient with the variables (a;j, ...2;„) of the forms, then any rational function of the coefficients of the forms and the coordinates of the points which is merely multiplied by the futh power (^jjL an integer) of the determinant of the transformation when the xs are subjected to any non-singular linear transformation is called a relative covariant of weight fx of the system of forms and points.* We may regard an invariant as the extreme case of a covariant where the number of points is zero. The other extreme case is that in which the number of forms is zero. Here we have the theorem: Theorem 3. The determinant pW ...4'»i is a relative covariant of weight — 1 of the system of points {x'„...x:),(4,:.x':),...{xf,...xi-^). For applying the transformation x^=c^^Xi+ ■■■ + Ci„Z„ ^n=<^nl^l + ••• + ''nn^t »' * In most books where the subject of covariants is treated, the same letters (xi, ■ ■ ■ Xn) are used for one of the points as for the variables of the forms. There is no objection to this, and it is sometimes convenient. We prefer to use a notation which shall make it perfectly clear that the variables of the forms have no connection with the coordinates of the points except that they are cogredient with them. 98 INTRODUCTION TO HIGHER ALGEBRA we have M ...M o,^X[ +-+c^„X^ -e^^X^ +-+c„„X', c,,X[-^ +■■■+ ^i„XW - c„iXiW + ... + c„„XN 'nl ^nn XW-XM XI -XL = eii---<''i,> <^nl'" "''ran -1 4"! ••■4" Or ?,s was to be proved. Another extremely simple case arises when we have a single form and a single point: Theorem 4. The system consisting of the form f (x^, ••• a;„) and the point (i/i, • • • «/„) has as an absolute covariant with regard to linear transformations . . . fiyv-'-yn)- For let us denote / more explicitly as J y^xi *2' " ■ ' ^v ' ' ' •"«)' where aj, a^, ■■■ are the coefficients of/. If the coefficients after the transformation are a[, a^, ••-, we have This l)eing true for all values of the a;'s, will be true if the 2;'s are replaced by the y's. But when this is done, the a;"s will be replaced by the ^''s, since the a;'s and y's. are cogredient. Accordingly fia'v 4' ••• ■ y'v ■■■y'n) =f{«v «2' - ; yv •••y»)' as was to be proved. INVARIANTS, FIRST PRINCIPLES AND ILLUSTRATIONS 99 The three examples of invariants and covariants which have been given in this section are all polynomials in the coefficients of the forms and the coordinates of the points. Such invariants we shall speak of as integral rational invariants and covariants.* Theorem 5. The weight of an integral rational invariant cannot be negative.-^ Let ftj, a^, ■■• ; Jj, Jgi--- ; ••• be the coefficients of the system of forms, and let c^j be the coefficients of the transformation. It is clear that the coefficients a\, a'^,---; J^, l'^,---; ••• after the transformation are polynomials in the a's, 5's, etc., and in the e^'s. Now let I be an integral rational invariant of weight /t, /(«;, 4, •■• ; &;, J^, ... ; ...) = c«J(ai, flg, •.. ; \, b^,--; •••). where c is the determinant of the transformation. Suppose now that /JL were negative, fj,= —v. Then (5) Clia',,-; 6i,-.; ■•• ) = ^K, - ; b„ - ; ■.■). This equality, like the preceding one, is known to hold for all values of the c^/s for which e ^0. Hence, since the expressions on both sides of the equality are polynomials in the a's, J's, ■•• and the Cj/s, ive infer, by an application of Theorem 5, § 2, that we really have an identity. Let us now assign to the a's, J's, •• any constant values such that J(aj,--- ; Jj, •■• ; ■■■)^0. Then I{a[,---; Jj, ■■• ; ■■•) will be a poly- nomial in the c^/s alone, which, from (5), cannot be identically zero. The identity (5) thus takes a form which states that the product of two polynomials in the e^/s is a constant, and since the first of these polynomials, c", is of higher degree than zero, this is impossible. We will agree in future to understand by the terms invariant and covariant, invariants or covariants (absolute, relative, or arithmetical) with regard to all non-singular linear transformations. If we wish to consider invariants or covariants with regard to other sets of trans- formations, for instance with regard to real linear transformationh this fact will be explicitly mentioned. * All rational invariants and covariants may be formed as the quotients of such an we integral and rational ; cf. Exercises 4, 5, § 78. * It cannot be zero either ; cf. Theorem 5, §79. 100 INTRODUCTION TO HIGHER ALGEBRA Finally, let us note the geometric meaning to be associated wit^ the invariants and covariants which have been mentioned in this section. We confine bur attention to the case of four variables. The vanishing of the resultant of four linear forms gives a necessary and sufficient condition that the four planes determined by setting the forms equal to zero meet in a point. The vanishing of the co- variant of Theorem 3 is a necessary and sufficient condition that the four points lie in a plane. The vanishing of the covariant of Theorem 4 is a necessary and sufficient condition that the point iVv Vv 2^3' y^ ^^® ^"^ *^® surface /= 0. It will be seen that in aU eases we are thus led to a projective property ; cf. §§80, 81. 32, Some Theorems Concerning Linear Forms. Theorem 1. Two systems of n linear forms in n variables are equivalent with regard to non-singular linear transformations if neithet resultant is zero. Let (a^^x^ + --- + a^„x„ (K^i+'-'+hn'^n (1) • • • • • (2) be the two systems, whose resultants, .i>ni^i+----tb„nX^ «11- ••«i. b = hi- -hn am- •«»» K- ■Kn are, by hypothesis, not zero. Applying the transformations CKj — a^^X^ + • • • + (l-in^n I *m — *Bl*l + ■ ■ ■ + '^nri^K ' x[=hiix^+ ■■■■k-b^^x^ . 4 = *„i»i+ ■■■+b„ to (1) and (2) respectively, they are both reduced to the normal form (3) Now, since neither a nor b is zero, the transformations a and b have inverses, which when applied to (3) carry it back into (1) and (2) respectively. Hence the transformation b~ia carries (1) into (2\ INVARIANTS. FIRST PRINCIPLES AND ILLUSTRATIONS 101 Theorem 2. A system of n linear forms in n variables has no integral rational invariants * other than constant multiples of powers of the resultant. Let (1) be the given system and a its resultant, and let e be the determinant of a non-singular linear transformation which carries (l)overinto ^a\,x[+ ... + a[^x'„, (4) . *ni^i "T" ■ • ■ + "^nn^n- If we call the resultant of (4) a', we have a' = ac. Let /(an, • • • «„„) be any integral rational invariant of the system (1) of weight fi, and write J'=J(ali, ...<„). Then /' = c^I. Now let us assume for a moment that a #0, and consider the special transformation which carries over (1) into the normal form (3). In this special case we have a' = 1; hence, as may also be seen directly, ac = 1. Calling the constant value which /' has in this particular case k, we have , „ t- _„ 7- or (5) /= har-. This equality, in which h is independent of the coefficients a^y, has been established so far merely for values of the a^'s for which a # 0. Since fj, is not negative (cf. Theorem 5, § 31), we can now infer that (5) is an identity, by making use of Theorem 5, § 2. Thus we >,ee that /is merely a constant multiple of a power of the resultant, as was to be proved. COROLLAKY. A System of m linear forms in n variables has no in- tegral rational invariants {other than constants) when m <_ n. For such an invariant would also be an integral rational invariant of the system of n linear forms obtained by adding n-m new forms to the given system ; and hence it would be a constant multiple of a * It has the arithmetical invariant mentioned in Theorem 2, § 30. 102 INTRODUCTION TO HIGHER ALGEBRA power of the resultant of this system. This power must be zero, and hence the invariant must be a mere constant, as otherwise it would involve the coefficients of the added forms, and hence would not be an invariant of the system of original forms. EXERCISES 1. Prove that if we have two systems of n + 1 linear forms in n variables whose matrices are both of rank n, a necessary and sufficient condition that these two systems be equivalent with regard to non-singular linear transformation is that the resultants of the forms of one set taken n at a time be proportional to th» resultants of the corresponding forms of the other set. 2. Generalize the preceding theorem. 3. Prove that every integral rational invariant of a system of m linear forms in H variables (m > n) is a homogeneous polynomial in the resultants of these forms taken n at a time. 4. State and prove the tlieorems analogous to the theorems of the present sec- tion, including the three preceding exercises, when the system of linear forms is re- placed by a system of points. 33. Cross-ratio and Harmonic Division. Let us consider any four distinct points on a line ' (1) {^v *i)' («2' ^2)' (2-3' h\ (^4' h)- We have seen, in § 31, Theorem 3, that each of the six determinants . x-^t^ — ajgCj, x-^tg — x^tj, x-^t^ — x^t-^, Xgt^ -^l^S' "^4 2 •^'2 4' ^2 3 '^3^2' is a covariant of weight — 1. The ratio of two of these determinants is therefore an absolute covariant, and we might be tempted, by analogy with the examples of absolute covariants in Exercise 1, § 28, to expect that it might have a geometric meaning. It will be readily seen, howe\er, that this is not the case, for the value of the ratio of two of the determinants (2) will be changed if the two coordinates of one of the four points are multiplied by the same constant, and this does not affect the position of the points. It is easy, however, to avoid this state of affairs by forming such an expression as the following:* /3-v (12 3 4^ — ^^1^2 ~ ^2^1) (^3^4 ~ ^4^3) (V3-«3*2)(«4*l-2^lO' * The reversal of sign of the second factor in the denominator is not essential, but ia customary for a reason which will presently be evident. INVARIANTS. FIRST PRINCIPLES AND ILLUSTRATIONS 103 which is also an absolute covariant of the four points (1), and is called their cross-ratio or anharmonic ratio. More accurately it is called the cross-ratio of these four points when taken in the order written in (1).* In order to determine the geometric meaning of the cross-ratio of four points, let us first suppose the four points to be finite so that ^1*2*3^4=^0. Dividing numerator and denominator of (1, 2, 3, 4) by this product, we find the following expression for the cross-ratio in " terms of the non-homogeneous coordinates X^ of the points, (4) (1, 1, 6, 4)- (^x,-X,)(X,-XS Finally, denoting the points by Pj, Pg' -^S' ^4? ^^ ™*y write C5) CI, A ^,4) p^pj p^p- p^pj p^p^ In words, this formula tells us that the cross-ratio of four finite points is the ratio of the ratio in which the second divides the first and third and the ratio in which the fourth divides the first and third ; and that it is also the ratio of the ratios in which the first and third divide the second and fourth. In this statement, it must be remembered that we have taken the ratio in which C divides the points J., B as AC I BC, so that the ratio is negative if G divides AB internally, positive if it divides it externally. If we agree that the point at infinity on a line shall be said to divide any two finite points A., B of this line in the ratio -1- 1 (and this is a natural convention since the more distant a point the more nearly does it divide AB in the ratio -f 1) it is readily jeen, by going back to formula (3), that the first statement following (5) still holds if the second or fourth point is at infinity, while the second statement holds if the first or third is at infinity. Thus we have in all cases a simple geometric interpretation of the cross-ratio of four distinct points. The special case in which four points Py, Pj, P3, P4 are so situ- ated that (1, 2, 3, 4) = — 1 is of peculiar importance. In this case we have (1, 2, 3, 4) = (1, 4, 3, 2) = (3, 2, 1, 4) = (3, 4, 1, 2) = (2, 1, 4, 3) = (2, 3, 4, 1) = (4, 1, 2, 3) = (4, 3, 2, 1)=-1. * If these four .points are taken in other orders, we get different cross-ratios 1 (1, 2, 4, S), (1, 4, 3, 2), etc. Cf. Exercise 1 at the end of this section 104 INTRODUCTION TO HIGHER ALGEBRA The relation is therefore merely a relation between the two pairs of points Pj, Pg and P^, P^ taken indifferently in either order, and we say that these two pairs of points divide each other harmonically. From the geometric meaning of cross-ratio, we see that, if all four points are finite, the pairs Pj, Pg and Pg, P^ divide each other har- monically when, and only when, Pg and P^ divide Pj, Pg internally and externally in the same ratio ; and also vsrhen, and only when, Pj and Pg divide Pg, P^ internally and externally in the same ratio. If Pg or P^ lies at infinity, the first of these statements alone has a meaning, while if Pj or Pg lies at infinity, it is the second statement to which we must confine ourselves. It is easily seen that the case in which three of the four points, say Pj^, Pg, Pg, coincide, while P^ is any point on the line, may be regarded as a limiting form of two pairs of points which separate one anotlier harmonically. It is convenient to include this case under the term harmonic division, and we will therefore lay down the definition: Definition. Two pairs of points Pj, Pg and Pg, P^ on a line are said to divide one another harmonically if they are distinct and their cross-ratio taken in the order Pj, Pg, Pg, P^ is — 1, and also if at least three of them coincide. It will be seen that the property of two pairs of points dividing each other harmonically is a projective property in space of one dimension. The most important applications of cross-ratio come in geometry of two, three, or more dimensions where the points are not determined as above by two coordinates (or one non-homogeneous coordinate), but by more. Suppose, for instance, we have four distinct finite points on a line in space of three dimensions. Let the points be Pj, Pg, §1, $g, and suppose the coordinates of Pj, Pg are {x^, i/j, gj, t-^ and (ajg, yg, ^g, ig) respectively. Then the coordinates of Q^, Q^ may be written (^1 + ^^^g,^! -f X«/2,Si -f-Xag,^! -1- \t^),{x^ + iix^,yi ■{- tiy2,z^+ tte^^t^-^- ixt^. Now, let (6) AxJrBy+az-\-I)t = be any plane through Q.^ but not through P^, and we have (,Ax^ + By^ + Cz^ + J)t,) + ^Ax^ + By 2 + Cz^ + Dt^) = 0, INVARIANTS. FIRST PRINCIPLES AND ILLUSTRATIONS 105 or, since P^ does not lie on (6), Axi + By I + (7gi + JXj^ ^ _ Changing to non-homogeneous coordinates, we have AX^ + BY^+O'Z^ + I) t^ AX^ + BY^+CZ^ + D~ t^' If PiMi and P^M^ are. the perpendiculars from Pj and P^ on the plane (6), we have PiQi^PiM^^ AX^ + BY^+OZ^ + B t^ P^Q, P^M^ AX^ + BY^+aZ^ + B~ ^t^' In exactly the same way we get Consequently ^iQi /PiQ%_'^ P^J P^^~J This is the cross-ratio of the four points taken in the order P^, Q^, It is readily seen that if one of the two points §j or Q^ lies at in- finity, all that is essential in the ahove reasoning remains valid, and the cross-ratio is still X//t. The case in which one of the two points Pj or P^ is at infinity may be reduced to the case just considered by writing for the coor- dinates of ^1 and §2, (1^1, 77^, ?i, Tj) and (^2, t]^, ^^, Tj). The coordi- nates of Pj and Pj are then (ll-^f2' ^l-)jlv ?l-^?2> ^l-^'-4 (?1 - ?2' '?! - '?2' fl - ?2' ^1 - ■^2)- Accordingly, from what has just been proved, we see that the cross-ratio of the four points taken in the order ^j, Pj, Q^, P^ is \/^i. But this change of order does not change the cross-ratio. Hence in all cases we have the result : 106 INTRODUCTION TO HIGHER ALGEBRA Theokem 1. The cross-ratio of the four distinct poinis •* 2 ^^2' Vv ^2' 2/' §2 (^1 + /^^2' ^1 + A'«/2' ^1 + ^^2' ^1 + M*2)> to^en m tAe order P^, Q^, P^, Q^, is X/zj,. From this theorem we easily deduce the further result : Theokem 2. The cross-ratio of four points on a line is invariant with regard to non-singular collineations of space.* For the four points Pj, P^, Qi, Q^ of Theorem 1 are carried over by a non-singular coUineation into the four points ■* 2 '■''2' 2^2' ^2' 2/' ei (zi + \4, y[ + xz/^, gj + x4, t; + X^), §2 (4 + /^4' ^i + W2' «i + ^^2' *i + /"4)» whose cross-ratio, when taken in the order Pj, Q'^, P'^, Q'^, is also X//*. Theorems similar to Theorems 1 and 2 hold in space of two, and in general in space of n, dimensions and may be proved in the same way. EXERCISES 1. Denote the six determinants (2) by (1,2), (1,3), (1,4), (3,4), (4,2), (2,3), and write ^ = (1, 2)(3,4), fi=(l,3)(4,2), C=(l,4)(2, 3). Prove that six, and only six, cross-ratios can be formed from four points by taking them in different orders, namely the negatives of the six ratios which can be formed from A, B, C taken two and two. 2. Prove that A-i-B + C = 0, and hence show that if X is one of the cross ratios of four points, the other five wUl be 1 1 . 1 X-1 X X' ^~^' i3x' ~r' x^i- * This also follows from Exercise 5, § 24, INVARIANTS. FIRST PRINCIPLES AND ILLUSTRATIONS 107 3. Prove that the six cross-ratios of four distinct points are all different from each other except in the following two cases : («) The ease of four harmonic points, where the values of the cross-ratios are - 1, 2, i- (/8) The case known as four equianharmonic points, in which the values of the cross-ratios are — J ± JV— 3. 4. Prove Theorem 2, § 24, by making use of the fact that the cross-ratio of four points on a line is unchanged by non-singular projective transformations of the line. 5. By the cross-ratio of four planes which meet in a line is understood the cross-ratio of the four points in which these planes are met by any line which does not meet their line of intersection. Justify this definition by proving that if the equations of the four planes are Pi = 0, Pl + >^P2 =0,P2= 0, Pi + y-P2 = (pj and ^2 homogeneous linear polynomials in x, y, z, t), the cross-ratio of the four points in which any line which does not meet the line of intersection of the planes is met by the planes is A//^. 6. Prove that the cross-ratio of four planes which meet in a line is invariant with regard to non-singular coUineations. 34. Plane-Coordinates and Contragredient Variables. If Mj, u^, Mj, u^ are constants, and x-^, z^, x^, x^ are the homogeneous coordinates of a point in space, the equation (1) Mja;! + M2«2 + %^3 + ^4*4 = represents a plane. Since the values of the m's determine the posi- tion of this plane, the m's may be regarded as coordinates of the plane. We will speak of them && plane-coordinates, just as the a;'s (each set of which determines a point) are called point-coordinates. And just as we speak of the point {x-^, x^, x^, x^) so we will speak of the plane (mj, u^, Mg, mJ. The tt's are evidently analogous to homogeneous coordinates in that if they be all multiplied by the same constant, the plane which they determine is not changed. Suppose now that we consider the x's as constants and allow the m's to vary, taking on all possible sets of values which, with the fixed set of values of the x's, satisfy (1). This equation will now repre- sent a family of planes, infinite in number, each one of which is de- termined by a particular set of values of the m's and all of which pass through the fixed point (xy x^, x^, x^. The equation (1) may there- fore be regarded as the equation of a point in plane-coordinates, since it is satisfied by the coordinates of a moving plane which envelops this point, just as when the x's vary and the m's are constant, it is 108 INTRODUCTION TO HIGHER ALGEBRA the equation of a plane in point-coordinates, since it is satisfied by the coordinates of a moving point whose locus is the plane.* In the same way, a homogeneous equation of degree higher than the first in the m's will be satisfied by the coordinates of a moving plane which will, in general, envelop a surface. The equation will then be called the equation of this surface in plane-coordinates, f Let us now subject space to the collineation c x'i = CijXj^ + Cf^x^ + Ci^Xg + Ci^x^ (i = 1, 2, 3, 4). We will assume that the determinant c of this transformation is not zero ; and we will denote the cofactors in this determinant by Cjy. Then the inverse of the transformation c may be written c-i Xi = ^x' + ^'x' + ^x' + ^x' {i = 1, 2, 3, 4). COG Substituting these expressions, we see that the plane (1) goes over into (2) u\x\ + M^a;^ + u'^x'^ + u\x'^ = 0, ^^ere r c r r d u[=^u^ + ^u^+ ^Mg + ^u^ (i = 1, 2, 3, 4). We thus see that the m's have also suffered a linear transformation, though a different one from the aj's, namely, the transformation whose matrix is the conjugate (cf . § 7, Definition 2) of c~^. This transforma- tion d of the plane-coordinates is merely another way of expressing the collineation which we have commonly expressed by the transfor- mation c of the point-coordinates. The two sets of variables x and u are called contragredi^t variables according to the following Definition 1. Two sets of n variables each are called contragre- dient if whenever one is subjected to a non-singular linear transformation, the other is subjected to the transformation which has as its matrix the conjugate of the inverse of the matrix of the first. * Similarly, in two dimensions, the equation UiXi + M2X2 + UsXn = represents a line in the point-coordinate (a;i, x^, Kg) if «i, Ma, % are constants, or a point in the line-coordinates («i, U2, Ms) if a^i, x^, x^ are constants, t An example of this will be found in § 53. INVARIANTS. FIRST PRINCIPLES AND ILLUSTRATIONS 109 Precisely the reasoning used above in the case of four variables establishes here also the theorem : Theoeem.* If the two sets of contragredient variables Xy ■•• x„ and, Mj, ■■• w„ are carried over hy a linear transformation into aij, ••• x'„ and u[, ■■■ wjj, then will go over into u[x[ + u'^x'^ + • • • + u^x'^. In connection with this subject of contragredient variables it is customary to introduce the conception of contravariants, just as the conception of covariants was introduced in connection with the sub- ject of cogredient variables. For this purpose we lay down the Definition 2. If we have a system of forms in {x^, ••■ x^ and also a number of sets of variables, (u'^,--- mJj), (mj', ••• m"), ■••, contra- gredient to the x's, any rational function of the w's and the coefficients of the forms which is unchanged by a non-singular linear transformation of the x's except for being multiplied by the fMthpou'er (/u. an integer) of the determinant of this transformation is called a contravariant of weight fi. Thus the theorem that the resultant of n linear forms in n variables is an invariant of weight 1 may, if we prefer, be stated in the form : If we have n sets of n variables each, (mJ, ••• wjj), •■• (^"', ••• m^"-'), each of which is contragredient to the variables (x-^, ■••'a;„), the de- terminant of the us, is a contravariant of weight l.f It will be seen that the conception of contravariant, though sometimes convenient, is unnecessary, since the contragredient vari- ables may always be regarded as the coefficients of linear forms, and, when so regarded, the contravariant is merely an invariant. Similarly, the still more general conception of mixed concomitants, in which, besides the coefficients of forms and the contragredient variables, certain sets of cogredient variables are involved, J reduces to the familiar conception of covariants if we regard the contra- gredient variables as coefficients of linear forms. * This is really a special theorem in the theory of bilinear forms. Cf. the next chapter. t For other examples of contravariants in which coefficients also occur, see Chap. XII. X An example is u\Xi + MzXa + ••• + ii„Xn, the theorem above stating that this is an absolute mixed concomitant. 110 INTRODUCTION TO HIGHER ALGEBRA 35. Line -Coordinates in Space. A line is determined by two points («/j, y^, y^, y^), (sj, z^, z^, z^ which lie on it. It is clear that these eight coordinates are not all necessary to determine the line ; and it will be seen presently that the following six quantities deter- mine the line completely, and may be used as line-coordinates. Fiv Pis^ Pu ' i'34' i'42' P2S> where (1) Pv = Zi Zj • In other words, the p's are the two-rowed determinants of the matrix Vi Vi Vs Vi ^1 ^2 ^3 ^4 except that the sign of the determinant obtained by striking out the first and third column has been changed. These six ^'s are not all zero if, as we assume, the two points y and z are distinct. These six p's are connected by the relation (2) PnPzi + P\zPi'i+ PiiPii = ^* as may be seen either directly or by expanding the vanishing determinant Vi by Laplace's method in terms of the minors of the first two rows. That the jo's may really be used as line- coordinates is shown by the following two theorems: Theorem 1. When a line is given, its line-coordinates py are com- pletely determined except for an arbitrary factor different from zero by which they may all be multiplied. The definition (1) of the ^'s shows that they may all be multi- plied by an arbitrary factor different from zero without affecting the position of the line, since the «/'s (and also the g's) may be multi- plied by such a factor without affecting the position of the point. 1/2 Vz Vi h h H Vi y% Vi * CI. Exercise 2, § 33. INVARIANTS. FIRST PRINCIPLES ANB ILLUSTRATIONS 111 In order to prove our theorem it is sufficient to show that, if instead of the two points used above for determining the p"s we use two other points of the line, the line-coordinates (^v ^2.-' ^s> ^ih n = Zi Z. thus determined will be proportional to thepy's. Since the points Y and Z are collinear with the distinct points y, z, they are linearly- dependent upon them and we may write Accordingly Zi^^yi^-h^Zi (i= 1,2, 3,4). '-1 ^-2 *1 n^2 y\ Vi. = -ff>i,-» where K^ 0, as T and Z are distinct points. Theorem 2. Any six constants py satisfying the relation (2) and not all zero are the line-coordinates of one, and only one, line. That they cannot be the coordinates of more than one line may be seen as follows: Suppose the pi/s to be the coordinates of a line, and take two distinct points y and z on the line. The coordinates of these points may then be so determined that relations (1) hold. Let us suppose, for definiteness, that Pi2¥=Q* Now, consider the point whose coordinates are Cj«/i+C2 3i. By assigning to Cj and c^ first the values — 3^ and y^, then the values — z^ and y^, we get the two points (3) (0, Pi2^ Pis, Pi^ (i?2i, 0, ^23, ^24). where, by definition, pij = — Pa- These two points are distinct, since for the first of them the first coordinate is zero and the second is not, while for the second the second coordinate is zero and the first is not. These points ac- cordingly determine the line, and since they, in turn, are deter- mined by the ^'s, we see that the line is uniquely determined by the ^'s. * By a slight modification of the fonnulse this proof will apply to the case in which any one of the p's is assumed different from zero. 112 INTRODUCTION TO HIGHER ALGEBRA It remains, then, merely to show that any set of Pi/s, not all zero, which satisfy (2) really determine a line. For this purpose we again assume p^^ =?^ * and consider the two points (3) which, as above, are distinct. The line determined by them has as its coordinates i'u' PnPia^ PiiPw - P1ZP42- PiiPiz^ P\-iP& PxiPiv I Using the relation (2), the fourth of these quantities reduces to PiiPsp ^'^ that, remembering that the coordinates of a line may be multiplied by any constant different from zero, we see that we really have a line whose coordinates are p^j. In a systematic study of three-dimensional geometry these line- coordinates play as important a part as the point- or plane-coordi- nates ; and in the allied algebraic theories we shall have to consider expressions having the invariant property, into which these line- coordinates enter just as point-coordinates occur in covariants and plane-coordinates in contravariants. We may, if we please, regard these expressions as ordinary covariants, since the line-coordinates are merely functions of the coordinates of two points, but the co- variants we get in this way are covariants of a special sort, since the coordinates of the two points occur only in the combinations (1). As an example, let us consider four points {Xi, 1/i, Zi, t,) {i = 1, 2, 3, 4). The determinant of these sixteen coordinates is, by Theorem 3, § 31, a covariant of weight — 1. Let us denote by p[j and p'^j the coordi- nates of the lines determined by the first two and the last two points respectively. Expanding the four-rowed determinant just referred to by Laplace's method according to the two-rowed determinants of the first two rows, we get (4) P'nP'L + P'iiPzi + P'lzP'L + ^^13^42 + P'uP'k + PuP'iz- This, then, is an expression having the invariant property and in- volving only line-coordinates. Siu'ce the vanishing of the four-rowed determinant from which we started gave the condition that the four points lie in a plane, it follows that the vanishing of (4) gives a necessary and sufficient condition that the two lines p' and p" lie in a plane, or, what amounts to the same thing, that they meet in a point. * By a slight modification of the formulae, this proof will apply to the case in which any one of the ^'s is assumed difierent from zero. INVARIANTS. FIRST PRINCIPLES AND ILLUSTRATIONS 113 EXERCISES 1. Prove that, if the point-coordinates are slibjected to the linear transfer X'i = CilXl + Ci2X2 + CisXs + CuXi (i = 1, 2, 3, 4), the line-coordinates will be subjected to the linear transformation Pii = (e.l«j-2 - Ci2Cfl)pi2 + (CiiCj-3 - CisCji) JOis + (caCji - CnCjj)pu + (CisCjt - C,iCjs)psi 2. A plane is determined by three points (2/i> 2/2, ys, 2/0, (.^u «2, zs, Zi), (wi, Wi, ws, 104). Prove that the three-rowed determinants of the matrix of these three points may be used as coordinates of this plane, and that these coordinates are not distinct from the plane-coordinates defined in § 34. 3. A line determined by two of its points may be called a ray, and the line- coordinates of the present section may therefore be called ray-coordinates. A line determined as the intersection of two planes may be called an axis. If («i, «2, «3, M4) and (vi, V2, vs, Vi) are two planes given by their plane-coordinates, discuss the axis-coordinates of their intersection, 912, ?i3, ?i4, qsi, 942, 52S> where qy = «<% — ujVi. 4. Prove that ray-coordinates and axis-coordinates are not essentially differ- ent by showing that, for any line, the q's, taken in the order written in Exercise 3, are proportional to the ^'s taken in the order Pu, Pi2, P23, P12, Pis, Pli- 5. A point is determined as the intersection of three planes («1, U2, «3, Ui), (i'l, V^, Vs, Vi), (Wl, 102, Ws, Wi). Prove that the three-rowed determinants of the matrix of these planes may be used as coordinates of this point, and that they do not differ from the ordinary point-coordinates. Hence, show that all covariants may be regarded as invariants. CHAPTER Vin BILINEAR FORMS 36. The Algebraic Theory. Before entering on the study of quadratic forms, which will form the subject of the next five chapters, we turn briefly to a very special type of quadratic form in 2 m varia- bles, known as a bilinear form, and which, as its name implies, forms a natural transition between linear and quadratic forms. DEFiNiTioisr 1. A polynomial in the 2n variables (a:^, ••■ a;„), (yi< • • • i/n) i's called a bilinear form if each of its terms is of the first degree in the xs and also of the first degree in the i/s. Thus, for w = 3, the most general bilinear form is ^U^l^l + «122^iy2 + «13^l2'8 + fltjl-^a^'l + '^wPil/'i + ^23-''2y3 3 . This may be denoted, for brevity, by %aifc^j ; and, in general, we may denote the bilinear form in 2 w variables by (1) The matrix Sa«r iptyj' a = *ii -^nl "In is called the matrix of the form (1) ; its determinant, the determinant of the form ; and its rank, the rank of the form.* A bilinear form is called singular when, and only when, its determinant is zero. * It should be noticed that the bilinear form is completely determined when its matrix is given, so there will be no confusion if we speak of the bilinear form a. If two bilinear forms have matrices ai and a2, their sum has the matrix ai-)-a2. 'The bilinear form whose matrix is aiaa is not the product of the two forms, but is sometimes spoken of as their symbolic product. 114 BILINEAR POKMS 115 Let US notice that the bilinear form (1) may be obtained by starting from the system of n linear forms in the ^'s of matrix a, multiplying them respectively by x^, x^, ■•■ x„, and adding them together. It can also be obtained by starting from the system of n linear forms in the x's whose matrix is the conjugate of a, multiplying them respectively by ^^ y^, ■■■ y„, and adding them together. Using the first of these two methods, we see (cf. Theorem 1, § 31) that if the ^'s are subjected to a linear transformation with matrix d, the bilinear form is carried over into a new bilinear form whose matrix is ad. Using the second of the above methods of building up the bilinear form from linear forms, we see that if the x's are subjected to a linear transformation with matrix c, we get a new bilinear form the conjugate of whose matrix is a'c, where accents are used to denote conjugate matrices. The matrix of the form itself is then (cf. Theorem 6, § 22) c'a.* Combining these two facts, we have Theokem 1. If, in the bilinear form (1) with matrix a, we subject the xs to a linear transformation with matrix c and the y's to a linear transformation with matrix d, we obtain a new bilinear form with matrix c'ad, where c' is the conjugate of c. Considering the determinants of these matrices, we may say : Theorem 2. The determinant of a bilinear form is multiplied by the product of the determinants of the transformations to which the x's and y''s are subjected, f We also infer from Theorem 1, in combination with Theorem 7, § 25, the important result : J Theorem 3. The rank of a bilinear form is an invariant with re- gard to non-singular linear transformations of the x's and y^s. Definition 2. A bilinear form whose matrix is symmetric it called a symmetric bilinear form. * These results may also be readily verified without referring to any earlier theorems. t This theorem tells us that the determinant of a bilinear form is, in a generalized sense, a relative invariant. Such invariants, where the given forms depend on several sets of variables, are known as combinants. t This result n">^ also be deduced from Theorem 2, § 30. 116 INTRODUCTION TO HIGHER ALGEBRA Theorem 4. A symmetric bilinear form remains symmetric if we subject the x's and the y's to the same linear transformation. For if c is the matrix of the transformation to which both the x'a and the «/'s are subjected, the matrix of the transformed form will, by Theorem 1, be c'ac. Remembering that a, being symmtetric, is its own conjugate, we see, by Theorem 6, § 22, that c'ac is its own conjugate. Hence the transformed form is symmetric. EXERCISES 1. Prove that a necessary and sufficient condition for the equivalence of two bilinear forms with regard to non-singular linear transformations of the a;'s and y'a is that they have the same rank. 2. Prove that a necessary and sufficient condition that it be possible to factor a bilinear form into the product of two linear forms is that its rank be zero or one. 3. Prove that every bilinear form of rank r can be reduced by non-singular linear transformations of the x's and y's to the normal form xiyi + Xiy2+ •■■ +Xryr- 4. Do the statements in the preceding exercises remain correct if we confine our attention to real bilinear forms and real linear transformations? 5. Prove that a necessary and sufficient condition that the form xiyi + X2yi+ ... +x„2/„ should be unchanged by linear transformations of the x's and of the y's is that these be contragredient transformations. 37. A Geometric Application. Let (x^, ajj, x^) and (j/i, yj, y,) be homogeneous coordinates of points in a plane, and let us consider the bilinear equation 3 (1) "la^iXtyj^O. If («/j, ygi ^s) is a fixed point P, then (1), being linear in the x's, is the equation of a straight line p. The only exception is when the coefficients of (1), regarded as a linear equation in the x's, are all zero, and this cannot happen if the determinant of the form is differ- ent from zero. Thus we see that the equation (1) causes one, and only one, line p to correspond to every point P of the plane, pro- vided the bilinear form in (1) is non-singular. Conversely, if (2) Ax^ + Bx^+Cx^^Q BILINEAR FORMS 117 IS a line;?, there is one, and only one, point F which corresponds to it by means of (1), provided the bilinear form in (1) is non-singular. For if P is the point (y^, y^, y^), the equation of the line corresponding to it is (1), and the necessary and sufficient condition that this line coincide witli (2) is a,,t/, + a,^, + a,^, = pA, «2l2/l + «22y2 + «232/3 = P^^ HlVx + «322/2 + ^^33^3 = P C'. where /> is a constant, not zero. For a given value of /o, this set of equations has one, and only one, solution (y^, y^, y^), since the deter- minant a is not zero, while a change in p merely changes all the y's in the same ratio. Hence, Theorem. If the bilinear equation (1) is non-sinffular, it establishes a one-to-one correspondence between the points and lines of the plane. This correspondence is called a correlation. EXERCISES 1. Discuss the singular correlations of the plane, considering separately the cases in which the rank of the bilinear form is 2 and 1. 2. Examine the corresponding equation in three dimensions, that is, the equa- tion obtained by equating to zero a bilinear form in which n= i, and discuss it for all possible suppositions as to the rank of the form. 3. Show that a necessary and sufficient condition for three or more lines, which correspond to three or more given points by means of a non-singular corre- lation, to be concurrent is that the points be collinear. 4. Show that the cross-ratio of any four concurrent lines is the same as that of the four points to which they correspond by means of a non-singular correlation. 5. Let P be any point in a plane and p the line corresponding to it by means of a non-singular correlation. Prove that a necessary and sufficient condition for the lines corresponding to the points of p to pass through P is that the bilinear iorm be symmetrical. 6. State and prove the corresponding theorem for points and planes in space of three dimensions, showing that here it is necessary and sufficient that the form •le symmetrical or skew-symmetrical.* * The correlation given by a symmetric bilinear equation is known as a reciproca- tion. By reference to the formulae of the next chapter, it will be readily seen that in this case every point corresponds, in the plane, to its polar with regard to a fixed conic ; in space, to its polar plane with regard to a fixed quadric surface. The skew-symmetric bilinear equation gives rise in the plane merely to a very special singular correlation, In space, however, it gives an important correlation which is in general nou-singulai and is known as a null-system. Cf. any treatment of line geometry, where, however, the subject is usually approached from another side. CHAPTER IX GEOMETRIC INTRODUCTION TO THE STUDY OF QUADRATIC FORMS 38. Quadric Surfaces and their Tangent Lines and Planes. If Xj^, a^g, 2^3 are homogeneous coordinates in a plane, we see, by reference to §4, that the equation of any conic may be written (J^ll^^j + 0^22^2 "t" *33^3"t" ^*12^1'*'2 "■" -'^13-^l-^3 < -'*23'''2^3 ^^ Similarly, in space of three dimensions, the equation of any quadric surface may be written ^11*1 + ^2*1 + *33^3 + *44^| + '^'^t'^1^2 + ^ajg^ja^g + 2a.j^a;j2;^ + 2as^x^x^ + 2.a^^x^x^ + la^^x^x^ = 0. This form may be made still more symmetrical if, besides the coefficients aj2' *i3. ^i4' *34' *42' ^23' '^^ introduce the six other con- stants a^p a^i, a^p a^g, a^^, a^^, defined by the general formula The equation of the quadric surface may then be written U/i-tX-t "T~ it-tnX-iXn ~r" ClinX-tXn ~T" ^lAX-tXj + a!21^2'*'l "■" '*22*2 ' ^23'^2'''3 >" ^24'^2"^4 + Clg^XgX-^ + 0^2,^3X2 + (tgs''^^ + ("342^32^4 + Cl^^X^X-^ -f~ ^^2*^4*^2 "T" ^43*^4*^3 "T~ ^44*^4 ^^ ^> or for greater brevity (1) S aijXiXj = 0. Definition 1. — The matrix of the sixteen a's taken in the order written above is called the matrix of the quadric surface (1), the deter- minant of this matrix is called the discriminant of the quadric surface, its rank is called the rank of the quadric surface, and if the discrimi- nant vanishes, the quadric surface is said to he singular. iia GEOMETRIC INTRODUCTION TO QUADRATIC FORMS 119 A fundamental problem is the following: If (y^, y^^ yg, y^ and (a^, Zj, gg, z^) are two points, in what points does the line yz meet the surface (1)? The coordinates of any point on yz, other than y, may be written (zj + \yi, 22 + X^a' ^3 + '^ys' ^4 + X^i). A necessary and sufficient condition for this point to lie on (1) is 4 ^a^^z, + \y>) (z,- + \y>f = 0, or expanded, ■(2) 'la^zfj + 2 X^tti^iZ^ + X^'laijy^j- = 0. If the point y does not lie on (1), this is a quadratic equation in \. To each root of this equation corresponds one point where the line meets the quadric. Thus we see that every line through a point y which does not lie on a quadric surface, meets this surface either in two, and only two, distinct points, or in only one point. On the other hand, if y does lie on (1), the equation (2) reduces to an equation of the first degree, provided la^yiZj^O. In this case, also, the line meets the surface in two, and only two, distinct points, viz., the point y and the point corresponding to the root of the equa- tion of the first degree (2). Finally, if "S^ai^y^j = 'La^^^j = 0, the first member of equation (2) reduces to a constant, so that (2) is either satisfied by no value of \, in which case the line meets the surface at the point y only, or by all values of \ (if 'Za^jZiZj- = 0), in which case every point on the line is also a point on the surface. Combining the preceding results we may say: Theorem 1. If a quadrio surface and a straight line are given, one of the following three cases must occur : (1) The line meets the quadric in two, and only two, points, in which case the line is called a secant. (2) The line meets the quadric in one, and only one, point, in which case it is called a tangent.* (3) Every point of the line is a point of the quadric. In this case the line is called a ruling of the quadric. f * We shall presently diatinguisb between true tangents and pseudo-tangents, t Also called a generator, because, as will presently appear, the whole surface may be generated by the motion of such a line. 120 INTRODUCTION TO HIGHER ALGEBRA That all these three cases are possible is shown by simple exam pies ; for instance, in the case of the surface the three coordinate ayes illustrate the three cases. We shall often find it convenient to say that a tangent line meets the quadric in two coincident points. From the proof we have given of Theorem 1, we can also infer the further result : Theorem 2. If («/j, y^-, y^, y^ is a point on the quadric (1), then if (3) ia,^x,yi = 0* every line through y is either a tangent or a ruling of (1), otherwise every line through y which lies in the plane (4) ^dio^iyj = is a tangent or ruling of(l), while every' other line through y is a secant. A theorem of fundamental importance, which follows immediately from this, is : Theorem 3. If there exists a point y on the quadric (1) such that the identity (3) is fulfilled, then (1) is a cone with y as a vertex ; and, con- versely, if (1) is a cone with y as a vertex, then the identity (2>)is fulfilled. We pass now to the subject of tangent planes, which we define as follows : Definition 2. A plane p is said to he tangent to the quadric (1) at one of its points. P, if every line of p which passes through P is either a tangent or a ruling of (1). It will be seen that, according to this definiti on, if (1) is a cone, every plane through a vertex of (1) is tangent to (1 ) at this vertex. We have thus included among the tangent planes, planes which in ordinary geometric parlance would not be called tangent. The same objection applies to our definition of tangent lines. We therefore now intro- duce the distinction between true tangent lines or planes and pseudo- tangent lines or planes. Definition 3. A line or plane which touches a quadric surface at a point which is not a vertex is called a true tangent ; all other tangent lines and planes are called pseudo-tangents. * It should be noticed that, on account of the relation a<; = o/c, 'Zaijx,0j = '2,(hi!]iXj. GEOMETRIC INTRODUCTION TO QUADRATIC FORMS 121 EXSRCISES 1. Prove that if P is a point on a quadrio surface S, which is not a vertex, and p the tangent plane at this point, one of the following three cases must occur : (a) Two, and only two, lines of p are rulings of S, and these rulings intersect at P. (i) One, and only one, line of /) is a ruling of S, and this ruling passes through P. , (c) Every line of p is a ruling of S. 2. Prove that (a) When case (a) of Exercise f occurs, the quadric surface is not a cone ; and, conversely, if the quadric surface is not a cone, case (o) will always occur. (b) If case (6) of Exercise 1 occurs, p is tangent to S at every point of the ruling which lies in p. (c) If case (J) of Exercise 1 occurs, S is a cone with one, and only one, vertex, and this vertex is on the ruling which lies in p ; and conversely, if S is a cone with one, and only one, vertex, case (J) will always occur. (d) If case (c) of Exercise 1 occurs, there is a line I iap every point of which (but no other point) is a vertex of S ; and S consists of two planes one of which is p, while the other intersects it in I. 39. Conjugate Points and Polar Planes. Two points are com- monly said to be conjugate with regard to a quadric surface 4 (1) ^a^jXiXj = 0, when they are divided harmonically by the points where the line connecting them meets the surface. In order to include all limiting cases, we frame the definition as follows: Definition. Two distinct points are said to be conjugate with re- gard to the surface (1) if (a) The line Joining them is a tangent or a secant to (1), and the points are divided harmonically hy the points where this line meets (1) ; or (h) The line joining them is a ruling of (1). Two coincident points are called conjugdte if they both lie on (1). Let the coordinates of the points be («/j, y^, y^, y^) and (gj, z^, z^, z^), and let us look first at the case in which the points are distinct and neither of them lies on (1), and in which the line connecting them is a secant of (1). The points of intersection of the line yz with (1) may therefore be written (si + ^,^1, ^2 + ^t^z' ^3 + ^i J/3' H + \ yd («' = 1» 2), 122 INTRODUCTION TO HIGHER ALGEBRA where X^ and X^ are the roots of Equation (2), § 38. A necessary and sufficient condition for harmonic division is that the cross-ratio Xj/Xg have the value —1 ; that is XiH-\2 = ; or, referring back to Equation (2), § 38, (2) ^atji/^,.=0. We leave it for the reader to show that in all other cases in which y and a are conjugate this relation (2) is fulfilled ; and that, conversely, whenever this condition is fulfilled, the points are conjugate. That is : Theorem 1. A necessary and sufficient condition that the points y, z he conjugate with regard to (1) is that (2) he fulfilled. This theorem enables us at once to write down the equation of the locus of the point x conjugate to a fixed point y, namely, (3) \a,p,y,=0. Except when the first member of this equation vanishes identically, this locus is therefore a plane called the polar plane of the point y. We saw in the last section that the first member of (3) vanishes identically when (1) is a cone and «/ is a vertex. This is the only ease in which it vanishes identically ; for, if y is any point, not a vertex, on a quadric surface, (3) represents the tangent plane at that point; while if y is not on (1), the first member of (3) can clearly not vanish identically, since it does not vanish when the x's are replaced by the y's. Hence the theorem : Theoebm 2. If (1) is not a cone, every point y has a definite polar plane (3) ; if (1) is a cone, every point except its vertices has a definite polar plane (3), while for the vertices the first member of (3) is identi- cally zero. We note that the property that a plane is the polar of a given point with regard to a quadric surface is a projective property, since a collineation of space evidently carries over two conjugate points into points conjugate with regard to the transformed surface. Theorem 3. If two points Pj and P^ are so situated that the polar plane of P.^ passes through P^, then, conversely, the polar plans of P, will pass through Pj. GEOMETRIC INTRODUCTION TO QUADRATIC FORMS 123 For, from the hypothesis, it follows that P^ and Pg are conjugate points, and from this the conclusion follows. 40. Classification of Quadric Surfaces by Means of their Rank. Theorem 2 of the last section may be stated by saying that a neces- sary and sufficient condition that the quadric surface be a cone and that (yj, y^, yj, t/^) be its vertex (or one of its vertices] is that (2) 'Laif>iyj = 0. This identity (2) is equivalent to the four equations ' «liyi + «122'2 +«132/3 + "-uVi = 0, (3) «2iyi + «22y2 + «23y3 + «242/4 = 0, «3l2/l + «322/2 + Hzyz+ ^34^4 = 0, • «4iyi + ^42^2 + «43y3+ «44y4 = 0- A necessary and sufficient condition for this set of equations to have a common solution other than (0, 0, 0, 0) is that the determi- nant of their coefficients be zero. We notice that this determinant is the discriminant a of the quadric surface. Hence, Theorem 1. A necessary and sufficient condition for a quadric surface to he a cone is that its discriminant vanish. If, then, the rank of the quadric surface is four, the surface is not a cone. If the rank is three, the set of equations (3) has one, and, except for multiples of this, only one, solution. Hence in this case the sur- face is an ordinary cone with a single vertex. If the rank is two, equations (3) have two linearly independent solu- tions (cf. §18), on which all other solutions are linearly dependent. Hence in this case the surface is a cone with a whole line of vertices. If the rank is one, equations (3) have three linearly independent solutions on which all other solutions are linearly dependent. Hence we have a cone with a whole plane of vertices. If the rank is zero we have, strictly speaking, no quadric surface ; but the locus of (1) may be regarded as a cone, every point in space being a vertex. 124 INTRODUCTION TO HIGHER ALGEBRA It is clear that the property of a quadric surface being a cone is a projective property ; and the same is true of the property of a point being a vertex of a cone. Hence from the classification we have just given we infer Theorem 2. The rank of a quadric surface is unchanged hy non- singular collineations. EXERCISES 1. Definition. If a plane p is the polar of a point P with regard to a quadric surface, then P is called a pole of p. Prove that if the quadric surface is non-siugular, every plane has one, and only one, pole. 2. Prove that if the quadric surface is a cone, a plane which does not pass through a vertex has no pole. What can be said here about planes which do pass through a vertex ? 41. Reduction of the Equation of a Quadric Surface to a Normal Form. Since cross-ratio is invariant under a non-singular coUinea- tion, a quadric surface S, a point P, not on S, and its polar plane with regard to S, are carried over by any non-singular collineation into a quadric surface S', a point P', and its polar plane with regard 4 to *S". A point (g^, y^, «/g, y^ not on the quadric surface '%aijXiX^= 0, cannot be on its own polar plane z^a^jX^j = as we see by replacing the x's in this last equation by the z/'s. Now transform by a colline- ation so that this point becomes the origin and its polar plane the plane at infinity.* The quadric surface will now be a central quad- ric with center at tlie origin, since, if any line be drawn through the origin, the two points in which this line meets the surface are divided harmonically by the origin and the point at infinity on this line. The equation of the polar plane of the point (y/, y'^, y'^, y^) with regard to the transformed quadric 4 'La'i-x'ix'j = 4 is '^a'iAy'i = 0, * Such a_j!ollineation can obviously be determined in an infinite number of ways by means of the theorem that there exists a collineation which carries over any five Unearly independent points into any five linearly independent points ; cf. Exercises 2. B, §24. GEOMETRIC INTRODUCTION TO QUADRATIC FORMS 125 ivhich reduces to the simple form aiiXi + a^ix'i + a'^xl + a'iix[ = when the point is the origin (0, 0, 0, 1), For this equation to rep- resent the plane at infinity, we must have Hence the quadric surface becomes «ii ^1 + «i2 ^i 4 + «i3 4 4 + a^i 4 x[ + «^2 42 + ^3 4 4 + «8i 4 4 + «32 4 4 + <3 4^ A slightly different reduction can be performed by transforming the point {y^, y^, 3/3, y^ to the point at infinity on the ajj-axis and its polar plane to the aiga^g-plane. It is easy to see that we thus get rid of the terms pontaining Xj except the square term. Similarly we can get rid of the terms containing x^ and x^ Thus we see that any quadric surface can he reduced hy a collineation to a form where its equation contains no term in x^ except the term in x} whose coefficient then is not zero. According as we take for i the values 1, 2, 3, 4, we get thus four different normal forms for the equation of our quadric surface, and inasmuch as each of these forms can be obtained in a great variety of ways, the question naturally arises whether we cannot perform all four reductions simultaneously. That this can, in general, be done may be seen as follows : let ^ be a point not on the quadric surface, and z any point on the polar plane of y, but not on the quadric surface. Its polar plane contains ^. Let w be any point on the intersection of the polar planes of y and s, but not on the quadric surface. Then its polar plane passes through y and S. These three polar planes meet in some point u, and it is readily seen that the four points y, z, w, u do not lie on a plane. The tetrahedron yzwu is called a polar or self- conjugate tetrahedron of the quadric surface, since it has the property that any vertex is the pole of the opposite face. If we transform the four points, y, z, w, u to the origin and the points at infinity on the three axes, the effect will be the same as that of the separate transformations above, that is, the equation of the quadric surface will be reduced to the form *11 "^1 ' ^22 "^2 ' ^33 ^S ' *14 ^i '^ 126 INTRODUCTION TO HIGHER ALGEBRA We have tacitly assumed that it is possible to find points y, z, w, constructed as indicated above, and not lying on the quadric surface. We leave it for the reader to show that, if the quadric surface is not a cone, this will always be possible in an infinite number of ways. A cone, however, has no self-conjugate tetrahedron, and in this case the above reduction is impossible. EXERCISES 1. Prove that if the discriminant of a quadric surface is zero, the equation of the surface can always be reduced, by a suitable collineation, to a form in which the coordinate x^ does not enter. [Suggestion. Show, by using the results of this chapter, that if the vertex of a quadric cone is at the origin, 014 = 024 = an = 044 = 0.] 2. Show that, provided the cone has a finite vertex, the collineation of Exercise 1 may be takeu in the form •Ci ^^ OC-i ~\' OL Xij ''^a ~ **'3 ")" y -^ii [Suggestion. Uee non-homogeneous coordinate,] CHAPTER X QUADRATIC FORMS 42. The General Quadratic Form and its Polar. The general quadratic form in n variables is n f 1) ^dtPiXj = ajja^ + ai22;i ajj + . . . + ai„XjX„ + OiiXiX-^ 4- a^ixl + . . . + a^^x^n T ''nl^n^ 4" (^ra-^n^2 + • • • + ^nn^'i S where a^ = ay^.* The bilinear form ^ai^yiZj is called the polar form of (1"). Subjecting (1) to the linear transformation we get a new quadratic form (2) ^c^x^r. The polar form of (2) is ^a^jylz'j. If we transform the ^'s and s's of the polar form of (1) by the same transformation c, we get a new n bilinear form '2a^-yiZj. We will now prove that a^ = a^. We have the identities n n (3) '^aijXiXj = l.aljxlx^, (4) 2aij2^i%=Saiyt/^sj. * It should be clearly understood that this restriction is a matter of convenience, Tjot of necessity. If it were not made, the quadratic form would be neither more noi less general. 127 128 INTRODUCTION TO HIGHER ALGEBRA Each of these we may regard as identities in the a;"s, y"s, z's, the x's, «/'s, z's being merely abbreviations for certain polynomials in the corresponding primed letters. The last written identity reduces, when we let ?/J = gj = cc^i = 1, 2, ... n), to ^ilyjXjJUj — itt^V^^X-n* • Combining this with (3) gives Hence a« = a'u and a,;,- + a^i = a^ + ajj. We have assumed that a'-, = ajj-, these being merely the coefficients of a certain quadratic form, and we proved, in Theorem 4, § 36, that a^- = ttji. Hence we infer that a^J = «,y. From this fact and from (4) we get at once the further result : n n That is : ^ * Theorem. The polar form is ar^hsolute eovariant of the system composed of the quadratic form n and the two points (y^, . . . y„), {z^, . . . z„). 43. The Matrix and the Discriminant of a Quadratic Form. Definition. The matrix a,, ... a. a = '11' is called the matrix of the quadratic form (1) 2^ai^-Xii The determinant of sl is called the discriminant of (1) ; and the rank of a, the rank of (1). If the discriminant vanishes, (1) is called singular. QUADRATIC FORMS 129 The matrix of (1) is the matrix of its polar form. Moreover, as was sliown in the last section, if the x's in (1) are subjected to a linear transformation, and the t/'s and z's in the polar of (1) are sub- jected to the same transformation, the matrix of the new quadratic form will be the same as the matrix of the new bilinear form. But we saw, in Theorem 1, §36, how the matrix of a bilinear form is changed by linear transformations of the variables. Thus we have the theorem : 'Thbokem 1. If in the quadratic form (1) with matrix a we sub- ject the x's to a linear transformation with matrix c, we obtain a new quadratic form with matrix c'ac, where d is the conjugate of c. From this there follow at once, precisely as in § 36, the further results : Theorem 2. The rank of a quadratic form is not changed by non- singular linear transformation. Theorem 3. The discriminant of a quadratic form is a relative invariant of weight two. 44. Vertices of Quadratic Forms. Definitiok. By a vertex of the quadratic form (1) ^aijXfOj, 1 we understand a point (cj, •••<;„) where the c's are not all zero, such that n (2) '^afjXiCj = 0. A quadratic form clearly vanishes at all of its vertices. It is merely another way of stating this definition when we say : Theorem 1. A necessary and sufficient condition that (cj, •■■ c„) be a vertex of (1) is that it be a solution, not consisting exclusively of zeros, <)f the system of equations (3) :::::::: 130 INTRODUCTION TO HIGHER ALGEBRA Since the resultant of (3) is the discriminant of (1), we may add: Theorem 2. A necessary and sufficient condition for a quadratic form to have a vertex is that its discriminant he zero ; and if the rank of the form is r, it has n—r linearly independent vertices, and every point linearly dependent on these is a vertex. In particular, we note that if the discriminant of a quadratic form is zero and if the cofactors of the elements of this determinant are denoted in the ordinary way by Ay, then (J.^, ■■•-4j„) is a vertex, provided all these ^'s are not zero. The following identify is of great importance (cf. formula (2), §38), (4) Sa<,- {Zi + \y,) {zj + X%) = 2 a^zfj + 2 Xla^ Ziyj + ^^^.a^yiyj. This may be regarded as an identity in all the letters involved. If (cj, • ■ • c„) is a vertex of the quadratic form 1,a^x,xj, and these c's are substituted in (4) in place of the ^'s, the last two terms of tiae second member of this identity are zero, and we have n n (5) Srty- (Si + \c,) {zj + Xcj) = S a^ZiZj ; and conversely, if (5) holds, (c^, •■• c„) is a vertex ; for subtracting (5) from (4), after substituting the c's for the ^'s in (4), we have n n 2 X^a^zfij + A,22 a^ c^c^ = 0, and, this being an identity in \ as well as in the z's, we have n 2«j,ZiC,- = 0. 1 Thus we have proved the following theorem: Theorem 3. A necessary and sufficient condition that (ffj, •••<?„) he a vertex of the quadratic form (1) is that 2j, ■••3„ and X being in- dependent variables, the identity (5) be fulfilled. EXERCISES 1. Prove that if (Cj, . . . c„) is a vertex of (1), and (y^, . . . y„) is any point at which the quadratic form vanishes, then the quadratic form vanishes at every point linearly dependent on c and y. 2. State and prove a converse to 1. QUADRATIC FORMS 131 45. Reduction of a Quadratic Form to a Sum of Squares. If in the quadratic form (1) ^(xj^,-x„) = 'S.ayXiXj the coefficient a^,- is not zero, we may simplify the form by the fol- lowing transformation due to Lagrange. The difference is evidently independent of Xf. Denoting it by ^j, we have ^ = ~ {ai^x^ + ■■■+ a,-„a;„)2 + (jty da If, then, we perform the non-singular linear transformation ■x{ = a,^x^ + a^^x^ + ■■■ +a,„a;. (2) Xo a/ = L « S/ft' the quadratic form <f> is reduced to the form (3) -^f + 0i(4»-^«). in which all the terms in x[ are wanting except the term in xf. It will be seen that this reduction can in general be performed in a variety of ways. It becomes impossible only when the coefficients of all the square terms in the original quadratic form are zero. Unless, in the new quadratic form <^j, the coefficients of all the square terms are zero, we can apply the same reduction to this form by subjecting the variables a/^, •■■ x'„ to a suitable non-singular linear transformation. This transformation may also be regarded as a non- singular linear transformation of all the x''s : (x[, x'^, • ■■ a;J,) if we write x'^ = x'y We thus reduce (3) to the form Applying this reduction now to (f)^, and proceeding as before, we see that by a number of successive non-singular transformations the form <f) can finally be reduced to the form : (5) 0^x1 + 6^x1+ ■■■ + e,x2. 132 INTRODUCTION TO HIGHER ALGEBRA These successive transformations can now be combined into a single non-singular linear transformation, and we are thus led to the Theorem. Every quadratic form in n variables can be reduced to the form (5) by a non-singular linear transformation. The proof of this theorem is not yet complete ; for if at any stage of the reduction the quadratic form ^^ has the peculiarity that all its square terms are wanting, the next step in the reduction will be impossible by the method we have used. Before considering this point, we will illustrate the method of reduction by a numerical case. Example. 12x1+ *i252 + 8a;j2;3"| + x^x^-ixl +9x^xA = X(^2x^ + X2 + 8xsy+<f)i + 8x^x^ + 9x^x^ + 2x1 J where f t o ^,^-l-ix,+ 8x,f-Bxl + 18x,x,+ 2xl^l^lll'^^^-^_g^^^^' =^-2(^-lx, + 5x,)^-l^xl. Accordingly, by means of the non-singular linear transformation (xj = 2x-^ + x2 +8 x^, ^2 = 2 ^2 ")~ " •*'S' ^3 = • Xg, the form <}> reduces to ^x'^ — ^x'^ — l^-S- x'^. We have given here merely owe method of reduction. Three differ- ent methods were open to us at the first step and two at the second, We proceed now to complete the proof of the general theorem, Let us suppose that the coefficients of all the square terms in </> ar» zero,* but that a-^2^^- Then ^(a^i, ■ ■ • a;„) s 2 a^^^^x^ + 2xj^ (a^^x^ + • ■ • + a^„x„) + 2 x^( a23^3 + • • • + «2na;„ ) + ^a^XiXj 2 ' = — (aja^s + «i3=»3 H 1- (^m^n) (a^v^i + H?Pz H V Hr^^r) + </>! 2 / " where ^i = (ajga^g +■■■+ a^^x^) (a^^x^ -\ h a^^x„) + ^Lai^x^Xj. «12 ^ ♦This method may be used whenever an = 022 = whether all the other coef ficients an are zero or net. QUADRATIC FORMS 133 Che non-singular linear transformation f *i = ai2*2 + «I3*3 + • • • + «!«*» + «2S*S + -" + «2na'» ^2 — ^21*1 La4 = thus reduces ^ to the form x^ — a;i'4 + ^j(a;3',-a;„'). "12 The further non-singular transformation Cj = ajj -f- x^, nf'' — fy' />»' 2 — 1 •''2' "8' r" = a!i reduces ^ to the form 1 2«12' 2 a. <2 + ^^(^«,...^^. 12 The above reduction was performed on the supposition that Ui^^f^O, It is clear, however, that only a slight change in notation would bn necessary to carry through a similar reduction if a-^^— ^ ^^^ '"'■a^ ^ The only case to which the reduction does not apply is, therefore, the one in which all the coefficients of the quadratic form are zero, a case in which no further reduction is necessary or possible. We thus see that whenever Lagrange's reduction fails, the method last explained will apply, and thus our theorem is completely eS' tablished. EXERCISES 1. Given a quadratic form in which n = 5 and a^ = \i —j\' Keduce to the form (5). 2. Keduce the quadratic form 9a;2_63/2-8z2 + 6a;y-14:a;z + 18a;tt; + 8y«H-12yio-4«w to the form C6> 1S4 INTRODUCTION TO HIGHER ALGEBRA. 3. Prove that if (y-^, ... y») is any point at which a given quadratic form is not zero, a linear transformation can be found (and that in an infinite number of ways) which carries this point into the point (0, ••• 0, 1) and its polar into kx„; and show that this linear transformation eliminates from the quadratic form all terms in x^ except the term in xj which then has a coefficient not zero. 4. Prove that the transformations described in Exercise 3 are the only ones which have the effect there described. 5. Show how the two methods of reduction explained in this section come as special cases under the transformation of Exercise 3. 46. A Normal Form, and the Equivalence of Quadratic Forms. In the method of reduction explained in the last section, it may happen that, after we have taken a number of steps, and thus reduced <f> to the form 0^x1+ ••• + c/^xl + 4>t (aji+i, ••■ x„), the form 0^ is identically zero. In this case no further reduction would be necessary and the form (5) of the last section to which <j) is reduced would have the peculiarity that c^+j while all the earlier e's are different from zero, when this case will occur. For this purpose, consider the matrix = 0, 0, It is easy to see just 'A+2 — ■ ■ ■ — ' of the reduced form (5) of § 45. It is clear that the rank of this matrix is precisely equal to the number of c's different from zero ; and, since the rank of this reduced form is the same as that of the original form, we have the result : Theorem 1. A necessary and sufficient condition that it be possi- ble to reduce a quadratic form by means of a non-singular linear trans- formation to the form (1) Cia;2+ •■• +c,x% where none of the c's are zero, is that the rank of the quadratic form be r. QUADRATIC FORMS 135 This form (1) involves r coefficients Cj, ••■ c^. That the values of these coefficients, apart from the fact that none of them are zero, are immaterial will be seen if we consider the effect on (1) of the trans- formation (2) C, = ->•' *r+l — ^< ■r+l» where \^ ••• Tc^ are arbitrarily given constants none of which, how- ever, is to be zero. The transformation (2) is non-singular, and reduces (1) to the form (3) \d^^ ••• +*X^. Thus we have proved Theokbm 2. A qtiadratio form of rank r can he reduced hy means of a non-singular linear transformation to the form (3), where the values of the constants k^, ■■■ k^ may be assigned at pleasure provided none of them are zero. If, in particular, we assign to all the ^'s the value 1, we get Theorem 3. livery quadratic form of rank r can he reduced to the normal form (4) xl-h ••• +x1. hy means of a non-singular linear transformation. From this follows Thbokem 4. A necessary and sufficient condition that two quad- ratic forms he equivalent with regard to non-singular linear trans- formations is that they have the same rank. That this is a necessary condition is evident from the fact that the rank is an invariant. That it is a sufficient condition follows from the fact that, if the ranks are the same, both forms can be reduced to the same normal form (4^). 136 INTRODUCTION TO HIGHER ALGEBRA The normal form (4) has no special advantage, except its sym- metry, over any other form which could be obtained from (3) by assigning to the Fs particular numerical values. Thus, for instance, a normal form which might be used in place of (4) is This form would have the advantage, in geometrical work, of giving rise to a real locus. Finally we note that the transformations used in this section are not necessarily real, even though the form we start with be real. EXERCISE! Apply the results of this section to the study of quadric surfaces. 47. Reducibility. A quadratic form is called reducible when it is identically equal to the product of two linear forms, that is, when n (1) 2 UijXiZj = (b^Xj^ + b^x^+--- + b„x„){ojXi +0^X2 -\ h c„x„). Let us seek a necessary and sufficient condition that this be the case. We begin by supposing the identity (1) to hold, and we consider in succession the case in which the two factors in the right-hand mem- ber of (1) are linearly independent, and that in which they are pro- portional. In the first case the 6's are not all proportional to the corresponding c's, and by a mere change of notation we may insure Jj, &2 not being proportional to Cj, c^. This being done, the trans- formation 'x[ = bjXi -1- J^a^a + • ■ • + Sn^« 4 = c^x^ + C2X^-\ h o„x^ is non-singular and carries our quadratic form over into the form The matrix of this form is readily seen to be of rank 2, hence the original form was of rank 2. QUADRATIC FORMS 137 Turning now to the case in which the two factors in (1) are proportional to each other, we see that (1) may be written n 2 ai^x,o;j = 0(bjX^ +■■■ + b„x^f where 0=^ 0. Unless all the 6's are zero (in which case the rank of the quadratic form is zero) we may without loss of generality suppose Jj^^O, in which ca' j the linear transformation a^ = Xo will be non-singular and will reduce the quadratic form to &'' which is of rank 1. ■^' Thus we have shown that if a quadratic form is reducible, its rank is 0, 1, or 2. We wish now, conversely, to prove that every quadratic form whose rank has one of these values is reducible. A quadratic form of rank zero is obviously reducible. A form of rank 1 can be reduced by a non-singular linear trans- formation to the form a;^, that is, If here we substitute for a^ its value in terms of the a;'s, it is clear that the form is reducible. A form of rank 2 can be reduced to the form x[ -1- x^^ that is, 2 a^iXi = x'' +x'^={x[ + ^/-l 4)(a4 - V^ 4). Here again, replacing a^ and a^ by their values in terms of the a;'s, the reducibility of the form follows. Hence, Theorem. A necessary and sufficient condition that a quadratic form be reducible is that its rank be not greater than 2. 48. Integral Rational Invariants of a Quadratic Form. We have seen that the discriminant a of a quadratic form is an invariant of weight 2. Any integral power of a, or more generally, any constant multiple of such a power, will therefore also be an invariant. We will now prove conversely the Theorem. Every integral rational invariant of a quadratic form is a constant multifile of some power of the discriminant. 138 INTRODUCTION TO HIGHER ALGEBRA Let us begin by assuming that the quadratic form n (1) '^aijXiXj is non-singular, and let o be the determinant of a linear transforma- tion which carries it over into the normal form (2) x'^ + x'i+-+x'i. Let /(aji, ••■ «„„)be any integral rational invariant of (l)of weight ii, and denote by k the value of this invariant when formed from (2). It is clear that A is a constant, that is, independent of the coefficients %of(l). Then h=c'--I. Moreover, the discriminant a being of weight 2, and having for (2) the value 1, we have ^ _ ^^_ Raising the last two equations to the powers 2 and /t respectively, we get p ^ ^^ j2^ 1 ^ ^^ ^^_ From which follows (3) J2 = Pa". This formula has been established so far for all values of the coefficients a^ for which a ^ 0. That it is really an identity in the a„'s is seen at once by a reference to Theorem 5, § 2. The poly- nomial on the right-hand side of (3) is of degree /x in aj, ; * hence we see that/i must be an even number, since P is of even degree in a^y Letting fjL='2v, we infer from (3) (cf . Exercise 1, § 2) that one or the other of the identities J= lca\ 1= - Tear must hold, and either of these identities establishes our theorem. A comparison of the result of this section with Theorem 4, § 46 will bring out clearly the essential difference between the two con- ceptions of a complete system of invariants mentioned in § 29. It will be seen that the rank of a quadratic form is in itself a complete sys- tem of invariants for this form in the sense of Definition 2, § 29 ; while the discriminant of the form is in itself a complete system in the sense of the footnote appended to this definition. * We assume here that k^O, .as otherwise the truth of the theorem would be obvious. QUADRATIC FORMS 139 49. A Second Method of Reducing a Quadratic Form to a Sum of Squares. By the side of Lagrange's method of reducing a quadratic form to a sum of squares, there are many other methods of accom- plishing the same result, one of the most useful of which we pro- ceed to explain. It depends on the following three theorems. The proof of the first of these theorems is due to Kronecker and estab- lishes, in a remarkably simple manner, the fact that any quadratic form of rank r can be written in terms of r variables only, a fact which has already been proved by another method in Theorem 1, § 46. Theorem 1. If the rank of the quadratic form n (1) (f>(x^,---x„) = '2aijXiXj is r > 0, and if the variables x^, ■•■ a;„ are so numbered that the r-rowed determinant in the upper left-hand corner of its matrix is not zero,* new variables x[, •■■ xl can be introduced by means of a non-singular linear transformation such that x[ = Xi (i = j- -Hi, •••»), and such that (1) reduces to the form r '^ayx'ix'j. This, it will be noticed, is a quadratic form in r variables in which the coefficients, so far as they go, are the same as in the given form (1). In order to prove this theorem, we begin by finding a vertex (cj, ••• c„) of the form (1) by means of Equations (3), § 44. Since the r-rowed determinant which stands in the upper left-hand corner of the matrix of these equations is not zero, the values of c^+i, ■■• c„ may be chosen at plfeasure, and the other c's are then completely deter- mined. If we let c^+i = c,+2 = •■• = c„_;i =0, c„ = 1, we get a vertex (cj, -e^O, -0, 1). Using this vertex in the identity (5), § 44, we have ^{x^-\-\o^, ■•• x^ + Xc^, Xr+i, •■•a;„_i,a;„-F\) = ^(a;i,-"a;„). If we let \ = — Xn, this identity reduces to (j) (xj — Cia;„, • • • a;^ — c^x„, x^^-^, • • • x„_i, 0) = ^ {x-y, • ■ • z„). * That such an arrangement is possible is evident from Theorem 3, § 20. 140 INTRODUCTION TO HIGHER ALGEBRA Accordingly, if we perform the non-singular linear transformation * \ = Xi-CiX„ (i=l, •••»•), [x'i = Xi (i = r + l,-n), the quadratic form (1) reduces to B-t <^(ai, -24-1,0)= 2a.ja^a^. This, being a quadratic form in n — 1 variables of rank r and so arranged that the r-rowed determinant which stands in the upper left-hand corner of its matrix is not zero, can be reduced, by the method just explained, to the form n-2 ZCl^/Xf Xj , where the linear transformation used is non-singular and such that 3^! = 3^i (i=r+l,-"n-l). By adding the formula x'l = a;J„ we may regard this as a non-singular linear transformation in the n variables. This transformation may then be combined with the one previously used, thus giving a non-singular transformation in which x'! = Xi, (i=»--l-l, •"«), and such that it reduces (1) to the form «-2 1 ' Proceeding in this way step by step, our theorem is at last proved. In the next two theorems we denote by Aij in the usual way the cofactor of a^- in the discriminant a of the quadratic form (1). Theorem 2. If J.„„ ^ 0, new variables x\, ■■• xl can he introduced by a non-singular transformation in such a way that and that (1) takes the form • This transformation should be compared with Exercise 2, § 41. QUADRATIC FORMS To prove this we consider the quadratic form 141 Its discriminant is *ii «i,»-i »ln *n-l, 1 ^^ a_ Hence by means of a non-singular transformation of the kind used in the last theorem, an essential point being that x^ = x„, we get M-l 'Lai.XiXi - -r~xl = ?a„a;l or n-l 2 a^j Xi Xj = E Uij x\ x'j + -^ x'^. Theorem 3. If new variables x[, ■■• xl can he introduced hy a non-singular transforma ■ tion in such a way that , , and that (1) takes the form V ' / , 2a ' , , ■i O'ij XiXj-\- — a;„ a;„_ J . ■' -^B, n-l Let us denote by B the determinant obtained by striking out the last two rows and columns of a. Then (cf. Corollary 3, § 11) we have -^»-l, B-l -"-n-l^r, (2) aB = A A B, B— 1 -*^Bfl = -Al„.^^Q. Consider, now, the quadratic form (3) '^aijXiZ 2a 142 INTRODLCTIQN TO HIGHER ALGEBRA Its discriminant is (4) *ii *l,o-l *ln "'n-l.l ■•• <"^B-l,n-l *B-1, n -^n, n-1 *Bl -^re, n-1 — » — -^n, m-1^ ^«, n-1 ^ -Dl -^ I ) -^», »-l -^n, n-1 Vja.„^ ;j_;i/ which has the value zero, as we see by making use of (2). Not only does the determinant (4) vanish, but its principal minors obtained by striking out its last row and column and its next to the last row and column are zero, being A„„ and ^„_j_„_j respectively. The minor obtained by striking out the last two rows and columns from (4) is B, and, by (2), this is not zero. Thus we see (cf. Theorem 1, § 20) that the determinant (4) is of rank n — 2. Hence, by Theorem 1, we can reduce (3) by a non-singular linear transformation in which *n-l — ^1 ■n-V -^n ai = a;„ to the form 2a *-w, n~\ Hence n-2 71, n—1 CoKOLLAKY. Under the conditions of Theorem 3, the quadratic form (1) can he reduced to the form n-2 2a«a;52;j + - 2a •W-i-^^2) "-n, n—\ hy a non- singular linear transformation. To see this we have merely first to perform the reduction of Theorem S, and then to follow this by the additional non-singular transformation <=x'! (j=l, 2, ...»-2> •''n-1 — ■*'re-l •'-ni X„ QUADRATIC FORMS 143 Haying thus established these three theorems, the method of reducing a quadratic form completely is obvious. If the form (1) is singular, we begin by reducing it by Theorem 1 to r where r is the rank of the form. Unless all the principal (r— 1)- rowed minors of the discriminant of this form are zero, the order of the variables a^, ••• x^ can be so arranged that the reduction af Theorem 2 is possible, a reduction which may be regarded as a non- singular linear transformation of all n variables. If all the princi- pal (r— 1) -rowed minors are zero, there will be at least one of the cofactors Afj which is not zero, and, by a suitable rearrangement of the order of the variables, this may be taken as A^^^^^. The reduc- tion of Theorem 3, Corollary, will then be possible. Proceeding in this way, we finally reach the result, precisely as in Theorem 1, § 46, that a quadratic form of rank r can always be reduced by a non- singular linear transformation to the form It may be noticed that the arrangement of the transformation of this section is in a certain sense precisely the reverse of that of §45, inasmuch as we here leave at each step the coefficients of the unre- duced part of the form unchanged, but change the variables which enter into this part ; while in §45 we change the coefficients of the (lureduced part, but leave the variables in it unchanged. CHAPTER XI REAL QUADRATIC FORMS 50. The Law of Inertia. We come now to the study of real quadratic forms and the effect produced on them by real linear transformations. We notice, here, to begin with, that the only operations involved in the last chapter are rational operations (i.e. addition, subtraction, multiplication, and division) with the single exception of the radicals which come into formula (2), § 46. In particular the reduction of § 45 (or the alternative reduction of § 49) involves only rational oper- ations. Consequently, since rational operations performed on real quantities give real results, we have Theorem 1. A real quadrdtic form of rank r can be reduced hy means of a real non-singular linear transformation to the form (1) c^x'^ + c^x'i+ ■■■ +c^'? where Oy, ••• e^ are real constayits none of which are zero. As we saw in the last chapter, this reduction can be performed in a variety of ways, and the values of the coefficients Cj, . ■ • c^ in the reduced form will be different for the different reductions. The signs of these coefficients, apart from the order in which they occur, will not depend on the particular reduction used, as is stated in the following im- portant theorem discovered independently by Jacobi and Sylvester and called by the latter the Law of Inertia of Quadratic Forms: Theorem 2. If a real quadratic form, of rank r is reduced hy two real non-singular linear transformations to the forms (1) and (2) k,x'i-^ + k^x'l^+ .■■+kX\ respectively, then the number of positive cs in{V) is equal to the number of positive k's in (2). In order to prove this, let us suppose that the z''s and a;"'s have been so numbered that the first fi of the c's and the first v of the jfc's are positive while all the remaining c's and k's are negative. Our 144 REAL QUADRATIC FORMS 145 theorem will be established if we can show that jj- = v. If this is not the case, one of the two integers i^ and v must be the greater, and it is merely a matter of notation to assume that fi^ v. We will prove that this assumption leads to a contradiction. If we regard the x"s and x"'s simply as abbreviations for certain linear forms in the x's, (1) and (2) are both of them identically equal to the original quadratic form, and hence to each other. This iden- tity may be written (3) c,x[^ +...+ c,x^^-\c,,,\xl\, =K142 Let us now consider the system of homogeneous linear equations in (2:1, •••a;„), (4) 4=0, ... x'^ = 0, < + i = 0, ... 4 = 0. We have here v + n—fi<n equations. Hence, by Theorem 3, Corollary 1, § 17, we can find a solution of these equations in which all the unknowns are not zero. ' Let (yj^, ••■ y„) be such a solution and denote by y'i, y" the values of a;^, x'l when the constants ^i, ■ - • y^ are substituted in them for the variables 2^, ... a;„. Substituting the ^'s for the a;'s in (3) gives oiy'^+ - +«.^;^= -\K^-,\y':ii IW- The expression on the left cannot be negative, and that on the right cannot be positive, hence they must both be zero; and this is pos- sible only if ,_ ,_Q But by (4) we also have y'^^-^ = •■• = yl = 0. That is, (yi, ■■■yn) is a solution, not composed exclusively of zeros, of the system of n homogeneous linear, equations in n unknowns, x[ = 0, 4=0, ...:c^=0. The determinant of these equations must therefore be zero, that is, the linear transformation which carries over the x's into the a;''s must be a singular transformation. We are here led to a contradiction, and our theorem is proved. 146 INTRODUCTION TO HIGHER ALGEBRA We can thus associate with every real quadratic form two in tegers P and N, namely, the number of positive and negative coeffi- cients respectively which we get when we reduce the form by any real non-singular linear transformation to the form (1). These two numbers are evidently arithmetical invariants of the quadratic form with regard to real non-singular linear transformations, since two real quadratic forms which can be transformed into one another by means of such a transformation can obviously be reduced to the same expression of form (1).* The two arithmetical invariants P and N which we have thus arrived at, and the arithmetical invariant r which we had before, are not independent since we have the relation (5) P + N=r. One of the invariants P and iV is therefore superfluous and either might be dispensed with. It is found more convenient, however, to use neither P nor N, but their difference, (6) s = P - iV, which is called the signature of the quadratic form. Definition. By the signature of a real quadratic form is under- stood the difference between the number of positive and the number of negative coefficients tvhich we obtain ivhen we reduce the form by any real non- singular linear transformation to the form (1). Since the integers P and iVused above were arithmetical invari- ants, their difference s will also be an arithmetical invariant. It should be noticed, however, that s is not necessarily a positive in- teger. We have thus proved Theorem 3. The signature of a quadratic form is an arithmetical invariant with regard to real non-singular linear transformations. EXERCISES 1. Prove that the rank r and the signature s of a quadratic form are either both even or both odd ; and that ^ , ' — r<s<r. 2. Prove that any two integers r and s (r positive or zero) satisfying the con- ditions of Exercise 1 may be the rank and signature respectively of a quadratic form. * P is sometimes called the index of inertia of the quadratic form. «ll«12 021022 ' ...Ar = ^n- •Oir On- .Orr REAL QUADRATIC FORMS 147 3. Prove that a necessary and sufficient condition that a real quadratic form of rank r and signature s be factorable into two real linear factors is that either r<2; or r = 2, s = 0. 4. A quadratic form of rank r shall be said to be regularly arranged (cf. § 20, Theorem 4) if the x's are so numbered that no two consecutive ^'s are zero in the set Af) — 1, ^j — Oji, A2 — and that Ar^O. Prove that if the form is real and any one of these 4's is zero, the two adjacent A's have opposite signs. [Suggestion. In this exercise and the following ones, the work of § 49 should he consulted.] 5. Prove that the signature of a regularly arranged real quadratic form is equal to the number of permanences minus the number of variations of sign in the sequence of the ^'s, if the ^'s which are zero are counted as positive or as nega- tive at pleasure. 6. Defining the expression sgn x (read signum x) by the equations sgna;= + 1 a;>0, sgna;= a; = 0, sgna;= — 1 a;<0, show that the signature of a regularly arranged real quadratic form of rank r is sgn (^0^1) + sgn (^1^2) + ••• + sgn {Ar-iAr). 51. Classification of Real Quadratic Forms. We saw in the last section that a real quadratic form has two invariants with regard to real non-singular linear transformations, — its rank and its signa- ture. The main result to be established in the present section (Theorem 2) is that these two invariants form a complete system. If in § 46 the c's and A's are real, the transformation (2) will be real when, but only when, each c has the same sign as the corre- sponding k. All that we can infer from the reasoning of that section now is, therefore, that if a real quadratic form of rank r can be reduced by a real non-singular linear transformation to the form it can also be reduced by a real nan-singular linear transformation to the form ^t^a^Z-f- ••• +Kx% lis INTRODUCTION TO HIGHER ALGEBRA where the Fs are arbitrarily giveu real constants, not zero, subject to the condition that each k has the same sign as the corresponding c. Using the letters P and N for the number of positive and negative e's respectively, the transformation can be so arranged that the first P e's are positive, the last iV negative. Accordingly the first P k's can be taken as + 1, the last iV as — 1. From equations (5) and (6) of § 50, we see that P and N may be expressed in terms of the rank and signature of the form by the formulae (1) P = r±i, N='^. Thus we have the theorem : Theorem 1. A real quadratic form of rank r and signature s can be reduced by a "eal non-singular linear transformation to the normal form (2) x\+--+xl-xl^^ xl where P is given by (1). We are now able to prove the fundamental theorem : Theorem 2. A necessary and sufficient condition that two real quadratic forms be equivalent with regard to real non-singular linear transformations is that they have the same rank and the same signature. That this is a necessary condition is evident from the invariance of rank and signature. That it is. sufficient follows from the fact that if the two forms have the same rank and signature, they can both be reduced to the same normal form (2). Definition. All real quadratic forms, equivalent with regard to real non-singular linear transformations to a given form, and therefore to each other, are said to f/^rm a class.* Thus, for instance, smce every real non-singular quadratic form in four variables can be reduced to one or the other of the five normal forms, (3) • This term may be used in a similar manner whenever the conception of equiva 'ence is involved. xl + xl + xl + xl x\ + ^2 + xl — xl x\ ^ X\ — xl — xl x\ — a;| — 4 — xl -x\ — ~2 x„ — xl — xl REAL QUADRATIC FORMS 149 we see that all such forms belong to one or the other of five classes characterized by the values s = 4, 2, 0, - 2, - 4, r = 4. If, however, as is the case in many problems in geometry, we are concerned not with quadratic forms, but with the equations obtained by equating these forms to zero, the number of classes to be distin- guished will be reduced by about one half, since two equations are the same if their first members differ merely in sign. Thus there are only three classes of non-singular quadric surfaces with real equations, whose normal forms are obtained by equating the first three of the forms (3) to zero. These equations written in non-homogeneous coordinates are X2 -I- r2 + Z2 = - 1, X^+ Y^ + Z^= 1, The first of these represents an imaginary sphere, the second a real sphere, and the third an unparted hyperboloid generated by the revolu- tion of a rectangular hyperbola about its conjugate axis. It may readily be proved that this last surface may also be generated by the revolution of either of the lines Y= 1 X =+ Z about the axis of Z. We may therefore say : Theorem 3. There are three, and only three, classes of non-singular quadric surfaces with real equations. In the first the surfaces are imag- inary ; in the second real, hut their rulings are imaginary ; in the third they are real, and the rulings through their real points are real* This classification is complete from the point of view we have adopted of regarding quadric surfaces as equivalent if one can be transformed into the other by a real non-singular coUineation. The more familiar classification does not adopt this projective view, but distinguishes in our second class between ellipsoids, biparted hyper- boloids, and elliptic paraboloids ;_ and in the third class between un- parted hyperboloids and hyperbolic paraboloids. * If, as here, we consider not real quadratic forms, but real homogeneous quadratic equations we must use, not s, but \s\ as an invariant. In place of \s\ we may use what is known as the characteristic of the quadratic form, that is the smaller of the two in- tegers P N This characteristic is simply \{i — |s|). 150 INTRODUCTION TO HIGHER ALGEBRA EXERCISES 1. Prove that there are J(n + 1) (ra + 2) classes of real quadratic forms in n variables. 2. Give a complete classification of singular quadric surfaces with real equa- tions from the point of view of the present section. 52. Definite and Indefinite Forms. Definition. By an indefinite quadratic form is understood a real quadratic form such that, when it is reduced to the normal form (2), § 51, hy a real non-singular linear transformation, both positive and neg- ative signs occur. All other real quadratic forms are called definite ; * and we distinguish between positive and negative definite forms accord- ing as the terms in the normal form are all positive or all negative. In other words, a real quadratic form of rank r and signature s is definite if « = ±r, otherwise it is indefinite. f The names definite and indefinite have been given on account of the following fundamental property : Theorem 1. An indefinite quadratic form is positive for some real values of the variables, negative for others. A positive definite form is positive or zero for all real values of the variables ; a negative definite form, negative or zero. , The part of this theorem which relates to definite forms follows directly from the definition. To prove the part concerning indefi- nite forms, suppose the form reduced by a real non-singular linear transformation to the normal form (1) x'^ + -+x'}-x%^ 42 Regarding the x"s as abbreviations for certain real linear forms in the x's, let us consider the system of n—P homogeneous linear equa- tions (2) rK^+i = 0, a;i>^2 = 0'•••^»=0• Since these equations are real, and their number is less than the number of unknowns, they have' a real solution not consisting * Some writers reserve the name definite for non-singular forms,, and call the singular definite forms semidefinite. f Otherwise stated, the condition for a definite form is that the characteristic he zero. Cf. the footnote to Theorem 3, § 61. REAL QUADRATIC FORMS 151 exclusively of zeros. Let (^/j, •■■«/„) be such a solution. This solution cannot satisfy all the equations (3) z[ = 0,-z'p=0, for equations (2) and (3) together form a system of n homogeneous linear equations in n unknowns whose determinant is not zero, since it is the determinant of the linear transformation which reduces the given quadratic form to the normal form (1). Accordingly, if we substitute («/j, ■ • • y„) for the variables (x^, • ■ • a„) in the given quad- ratic form, this form will have a positive value, as we see from the - reduced form (1). Similarly, by choosing for the x's a real solution of the equations 0/^ = 0, ■■■x'p=0, 4+1 = 0, ■•• 2;^= 0, which does not consist exclusively of zeros, we see that the quad- ratic form takes on a negative value. We pass now to some theorems which will be better appreciated by the reader if he considers their geometrical meaning in the case w = 4. Theorem 2. If an indefinite quadratic form is positive at the real point (?/j, •■■ y„) and negative ai the real point (^^y, ■■■ s„), then there are two real points linearly dependent on these two, hut linearly inde- pendent of each other, at which the quadratic form is zero, and neither of which is a vertex of the form. The condition that the quadratic form n (4) ^ayXiXj vanish at the point (yj + XSj, ■••«/„ + Xs„) is This quadratic equation in \ has two real distinct roots, since, from our hypothesis that (4) is positive at y and negative at z, it follows that Let us call these roots \^ and \. Then the points (5) (^1 + XiSj, •••«/„ + \Zn\ (2/1 + Vi' - y« + ^'.2.) 152 INTRODUCTION TO HIGHER ALGEBRA are two real points linearly dependent on the points y and z at which (4) vanishes. Next notice that (6) IXj 1x3 Zi Zj Since the points y and z are linearly independent, the integers i,j can be so chosen that the last determinant on the right of (6) is not zero. Then the determinant on the left of (6) is not zero; and, consequently, the points (5) are linearly independent. In order, finally, to prove that neither of the points (5) is a " vertex, denote them for brevity by (Fj, ... F„), (^1, ... Z„). Letting Xj — X^ = l//i, we have Zi = ixTi - fiZi (i = 1, 2, ... n). Therefore (7) ia,,.a,3,. = ^fi^a,J YiY,- - 2 lAa^^Y^Z^ + ,J?la,^Z,Z^. Since the points Y and Z have been so determined that (4) vanishes at them, the first and last terms on the right of (7) are zero. If either Y ov Z were a vertex, the middle term would also be zero ; but this is impossible since the left-hand member of (7) is, by hypothesis, negative. Thus our theorem is proved. For the sake of completeness we add the corollary, whose truth is at once evident : Corollary. The only points linearly dependent on y and z at which the quadratic form vanishes are points linearly dependent on one or the other of the points referred to in the theorem; and none of these are vertices. We come now to a theorem of fundamental importance in the theory of quadratic forms. Theorem 3. A necessary and sufficient condition that a real quadratic form he definite is that it vanish at no real points except its vertices and the point (0, 0, ••• 0). REAL QUADRATIC FORMS 153 Suppose, first, that we have a real quadratic form which vanishes at no real points except its vertices and the point (0, 0, ••■ 0). If it were indefinite, we could (Theorem 1) find two real points «/, z, at one of which it is positive, at the other, negative. Hence (Theorem 2) we could find two real points linearly dependent on y and z, at which the quadratic form vanishes. Neither of these will be the point (0, 0, ■•• 0), since, by Theorem 2, they are linearly independ- ent. Moreover, they are neither of them vertices. Thus we see that the form must be definite, and the sufficiency of the condition is established. It remains to be proved that a definite form can vanish only at its vertices and at the point (0, 0, ••• 0). Suppose (4) is definite and that {y^^ •■■ Vn) is any real point at which it vanishes. Then, 'laijixi + >.?/i)(2j- + Xy,-) = 'la^-XiXj + 2 Xlaif^^yj. If y were neither a vertex nor the point (0, 0, ••■ 0), 'I,a{jX,yj would not vanish identically, and we could find a real point (sj, • • • z„) such that n k = ^a,jZit/j ^ 0. n If we let c = SaySjZ,-, we have (8) |«.i<2< + X«/i)(% + Xy,) = e+2\k. For a given real value of X, the left-hand side of this equation, is simply the value of the quadratic form (4) at a certain real point. Accordingly, for different values of X it will not change sign, while the right-hand side of (8) has opposite signs for large positive and large negative values of X. Thus the assumption that y was neither a vertex nor the point (0, 0, ■•• 0) has led to a contradiction; and our theorem is proved. Corollary. A non-gingular definite quadratic form vanishes, for real values of the variables, only when its variables are all zero. As a simple application of the last corollary we will prove Theorem 4. In a non-singular definite form, none of the coeffi. cients of the square terms can be zero- 164 INTRODUCTION TO HIGHER ALGEBRA For suppose the form (4) were definite and non-singular; and that Uii = 0. Then the form would vanish at the point a;^ = . . . = a;j_ J = Xi^-^ = • • • = a;„ ^ U, a;^ = 1 ; and this is impossible, since this is not the point (0,0, •••0). EXERCISES 1. Definition. By an orthogonal transformation * is understood a linear trans- formation which carries over the variables {xi,--- x„) into the variables ^x[,"- af„) in such a way that Xl + Xl+ ... + Xl = x[^ + Mi' + ... + X'^. Prove that every orthogonal transformation is non-singular, and, in particular, that its determinant must have the value + 1 or — 1. 2. Prove that all orthogonal transformations in n variables form a group; and that the same is true of all orthogonal transformations in n variables of deter- minant + 1. 3. Prove that a necessary and sufficient condition that a linear transforma- tion be orthogonal is that it leave the " distance " y/(.yi - ziy + iVi - ^Y + - + (yn - 2n)' between every pair of points (yi, ... y^, (zi, •.• z„) invariant. 4. Prove that if n = 3, and if x\, x^, xa be interpreted as non-homogeneous rectanglar coordinates in space, an orthogonal transformation represents either a rigid displacement which leaves the origin fixed, or such a displacement combined ■with reflection in a plane through the origin. Show that the first of these cases will occur when the determinant of the transformation is + 1, the second when this determinant is — 1. 5. If the coefficients of a linear transformation are denoted in the usual way by Cjj, prove that a necessary and sufficient condition that the transformation be orthogonal is that cl + c!i+ ... +cli=l (J = 1,2, ...n), „ ft = 1,2, ••• n . CliCy + CiiCgj + ... + CnlCnS = U 1 . , „ » ^J" U=1.2, ".n Show that these will still be necessary and sufficient conditions for an orthogona) transformation if the two subscripts of every c be interchanged.! * The matrix of such a transformation is called an orthogonal matrix, and its deter- minant an orthogonal determinant. t We have here J » (n + 1) relations between the n^ coefficients of the transforma- tion. This suggests that it should be possible to express all the coefficients in terms of j^3 wCw-f l) _ «(n-l) 2 2 of them, or if we prefer in terms of J n (m — 1) other parameters. Tor Cayley's dls- cifesion of this question cf. Pascal's book, Die Determinanten,. § 47. Cayley's formulse, however, do not include all orthogonal transformations except as limiting cases. CHAPTER XII THE SYSTEM OF A QUADRATIC FORM AND ONE OR MORE LINEAR FORMS 53. Relations of Planes and Lines to a Quadric Surface. If thb plane (1) UjX-^ + U^X^ + U^Xg + u^x^ = 9 is a true tangent plane to the quadric surface 4 (2) 2ai,a;,.a;y = 0, there will be a point («/j, y^, t/g, y^ (namely the point of contact) lying in (1) and such that its polar plane (3) •iaijXiyj=0 coincides with (1). From elementary analytic geometry we know that a necessary and sufficient condition that two equations of the first degree represent the same .plane is that their coefficients be pro- portional. Accordingly, from the coincidence of (1) and (3), we deduce the equations ^ii^i + «123'2 + '^izVz + "'uVi - /***! = ^' «2iyi + <*222'2 + S32/3 + ^24^4 " P'^2 = ^' «3iyi + H2^i + H^yz + ^34^4 - PW3 = ^■> . HxVl + ^42^2 + H^Z + «442/4 ^ /'**4 = ^• From the fact that the point y lies on (1), we infer the further relation (5) u-^yy + -1*2^2 + ^3^8 + ^42/4 = ^• These equations (4) and (5) have been deduced on the suppo- sition that (1) is a true tangent plane to (2). They still hold if it is a pseudo-tangent plane ; for then the quadric must be a cone, and a vertex of this cone must lie on (1). Taking the point y as this vertex, equation (5) is fulfilled. Moreover, since now the first 166 (4) 156 INTRODUCTION TO HIGHER ALGEBRA member of (3) is identically zero, equations (4) will also be fulfilled if we let /3 = 0. Thus we have shown in all cases, that if (1) is a tangent plane to (2), there exist five constants, ^j, y^, y^, y^, p, of which the first four are not all zero, and which satisfy equations (4) and (5). Hence (6) «u "-n «18 ai4 «1 «21 «22 «23 «24 Ma «31 «32 «33 «34 M3 «41 «42 «4S «44 M4 Ml "2 Ms M4 = 0. Conversely, if this last equation is fulfilled, there exist five constants, yj, y^, y^, y^, p, not all zero, and which satisfy equations (4) and (5). We can go a step farther and say that y^, y^., y^, y^ cannot all be zero, as otherwise, from equations (4) and the fact that the m's are not all zero, p would also be zero. Thus we see that if equation (6) is fulfilled, there exists a point {y^, y^, y^, t/4) in the plane (1) whose coordinates, together with a certain constant p, satisfy (4). If p = 0, this shows that the quadric is a cone with y as a vertex, and hence that (1) is at least a pseudo-tangent plane. If p =?^ 0, equatijons (4) show us that the polar plane (3) of y coin- cides with the plane (1). Moreover we see, either geometrically, or by multiplying equations (4) by ^j, y^, y^, y^ respectively and add- ing, that the point y lies on the quadric ; so that, in this case, (1) is a true tangent plane. We have thus established the theorem : Theorem 1. Equation (6) is a necessary and sufficient condition that the plane (1) he tangent to the quadric (2). It will be seen that this theorem gives us no means of distinguish- ing between true and pseudo-tangent planes of quadric cones. In the case of non-singular quadrics, pseudo-tangent planes are impos- sible, and therefore equation (6) may, in this case, be regarded as the equation of the quadric in plane-coordinates. In the case of a quadric surface of rank 3, that is, of a cone with a single vertex, the coordinates (mj, Mj, %, u^ of every plane through this vertex satisfy equation (6), so that in this case this equation represents a single point, and not the quadric cone.* * In fact a cone cannot be represented by a single equation in plane-coordinates. QUADRATIC AND LINEAR FORMS 157 If the rank of (2) is less than 3, the coordinates of every plane in space should satisfy (6), since every such plane passes through a vertex and is therefore a tangent plane. This fact may be verified by noticing that equation (6) may also be written 4 ZiA^jUflij = 0, where the J.'s are the cofactors in the discriminant of (2) according to our usual notation. We pass now to the condition that a straight line touch the quadric (2). This line we will determine as the intersection of the two planes (1) and (7) v^x^ + v^x^ + v^x^ + v^x^ = 0. If the line of intersection of these planes is a true tangent to (2), there will be a point (t/^, y^^ t/g, yj, namely the point of contact, lying upon it, and such that its polar plane (3) contains the line. It must there- fore be possible to write the equation of this polar plane in the form (8) i(fiUt + vVi)Xi=0; and, in fact, by properly choosing the constants fi and v, the co- efficients of (8) may be made not merely proportional, but equal to the coeiBcients of (3) : • «ll2/l + «12^2 + «132/3 + ^14^4 - f^^i - i'«'i= 0^ «21^1 + «22^2 + «23y3 + «242/4 " '^'^2 " ^2= <^' «3iyi + '^322/2 + «332/3 + «342/4- ^^3 " ''^S = 0, . «4iyi + «422/2 + «432/3 + <^id/i " f^^i " ^^i = ^' Since the point 1/ lies on the line of intersection of the planes (1) and (7), we also have the relations f Ml^l + W2«/2 + ^3^3 + «42^4 = 0' I Vi^l + ^23^2 + "3^3 + ^d/i = 0- Since the six equations (9) and (10) are satisfied by six constants ?/v Vt ^3' yv i^-' ^ ^°* ^ ^®'^°' ^® ^°^®'' ^"^^^^y ^^ relation (9) (10) ai) *31 "12 ^32 ^42 *13 ^23 ^33 ^43 *14 *24 -^34 ^^44 M, Mg = 0. 158 INTRODUCTION TO HIGHER ALGEBRA We have deduced this equation on the supposition that the line of intersection of (1) and (7) is a true tangent to (2). We leave it to the reader to show that (11) holds if this line is a pseudo-tangent, and also if it is a ruling of (2). We also leave it for him to show that if (11) holds, the line of intersection of (1) and (7) will be either a true tangent, a pseudo- tangent, or a ruling, and thus to establish the theorem: Theokem 2. A necessary and sufficient condition that the line of intersection of the planes (1) and (7) he either a tangent or a ruling of (2) is that equation (11) be fulfilled. On expanding the determinant in (11), it will be seen that it is a quadratic form in the six line-coordinates q^ (cf. Exercise 3, § 35). Equation (11) may therefore be regarded as the equation of the quadric surface in line-coordinates if the surface is not a cone, or is a cone with a single vertex. If the rank of (2) is 2, so that the. quadric consists of two planes, (11) is the equation of the line of intersection of these planes. While if the rank is 1 or 0, (11) is identically fulfilled. EXERCISES 1. Two planes are said to be conjugate with regard to a non-singular quadric surface if each passes through the pole of the other. Prove that if (2) is "- non-singular quadric, a necessary and sufficient con- dition that the planes (1) and (7) be conjugate with regard to it is the vanishing of the determinant "1 "12 Qia ' ^22 ^23 ^24 ^2 ^,32 ^aa ^M ^3 ^^42 "m "44 "4 [', V, V. = - ^AijViiVj. How must this definition of conjugate planes be extended in order that this theorem be trUe for singular quadrics also? 2. Prove that if (2) is a non-singular quadric, a necessary and sufficient con- dition that the point of intersection of three planes lie on (2) is the vanishing of the seven-rowed determinant formed by bordering the discriminant of (2) with the coefficients of the three planes. 3. Admitting it to be obvious geometrically that a necessary and sufficient con- dition that a line touch a non-singular quadric is that the two tangent planes which can be passed through this line should coincide, prove that, if (2) is non-singular, a necessary and sufficient condition that the line of intersection of (1) and (7) touch ^2^ is ^ ■* 4 (S AijUiUj) (2 Aig-ViVj) - (2 AijttiVjy = 0. 4. Show algebraically that the condition of Exercise 3 is equivalent to (11). QUADRATIC AND LINEAR FORMS 159 54. The Adjoint Quadratic Form and Other Invariants. Passing now to the case of n variables, we begin by considering the system consisting of a quadratic form and a single linear form (2) ^UiX{. The geometrical considerations of the last section suggest that we form the expression „ .._ ^ n , . . . (3) '2Aij.UiUj = - This, it will be seen, is a quadratic form in the variables (mi, ■■■u„) whose matrix is the adjoint of the matrix of (1). We will speak of (3) as the adjoint of (1). The invariance of (3) is at once suggested by the fact that in the case n = 4: the vanishing of (3) gave a necessary and sufficient con- dition for a projective relation. In fact we will prove the theorem : # Theorem 1. 2%e adjoint form (3) is an invariant of weight two of the pair of forms (1), (2). Inasmuch as the m's are, as we saw in § 34, contragredient to the x's, we may also call (3) a contravariant (cf. Definition 2, § 34). In order to prove this theorem we must subject the a;'s to a linear transformation, (4) Xi — Cjj a;i + • ■ • + Ci„2;„ whose determinant we will call c. Let us denote by a!y and i4 re- spectively the coefficients of the quadratic and linear form into which this transformation carries (1) and (2). Let us now introduce an auxiliary variable t, and consider the quadratic form in a^, ••• x„, t, .(5) '2aijXiXj+2t{iiiXi-\ f- u„ «;„). The discriminant of this form is precisely the determinant in (3), that is, the negative of the adjoint of (1). 160 INTRODUCTION TO HIGHER ALGEBRA Let us now perform on the variables x^, ••■x„,t the linear trana- formation given by formulae (4) and the additional formula (6) t=t'. The determinant of this transformation is c, and it carries over the form (5) into ",,,„,/,, , ,. From the fact that the discriminant of (5) is an invariant of weight 2, we infer the relation we wished to obtain : "11 *ln «»1 -^nl The method just used admits of immediate extension to the proof of the following more general theorem : Theorem 2. If we have a system consisting of a quadratic form in n variables and p linear forms, the in + p)-rowed determinant formed hy bordering the discriminant of the quadratic form by p rows and p columns each of which consists of the coefficients of one of the linea" forms is an invariant of weight 2. We leave the details of the proof of this theorem to the reader. If the discriminant a of the quadratic form (1) is not zero, we may form a new quadratic form whose matrix is the inverse of the matrix of (1). This quadratic form, which is known as the inverse or reciprocal of (i), is simply the adjoint of (1) divided by the discrimi- nant a. We will prove the following theorem concerning it : Theorem 3. If the quadratic form (1) is non-singular, it will be carried over into its inverse by the non-singular transformation (7) x'i = aii«i + ••• + ai^x„ (i= 1, 2, - «). For we have 2i a^j x^ Xj — 2j x^x^. But from (7) we have a a and theuefore as was to be proved. 2 aijXiXj = 2 — ^' x'ips'j. QUADRATIC AND LINEAR FORMS 161 It will be noticed that if (1) is a real quadratic form, the trans- formation (7) is real ; and from this follows Theoeem 4. A real non-singular quadratic form and its inverse have the same signature. EXERCISES 1. Given a quadratic form ^oyXiXj and two linear forms S««a;i, 2iViXi. Prove that 'S,AijUiVj=- iii ■•• Uin «, Uj ••• Vn is an invariant of the system of weight 2. 2. Generalize the theorem of Exercise 1 to the case in which we have more than two linear forms. 3. Prove that if a first quadratic form is transformed into a second by the linear transformation of matrix c, then the adjoint of the first will be transformed into the adjoint of the second by the linear transformation whose matrix is the conjugate of the adjoint of c. 4. Prove a similar theorem for bilinear forms. 5. State and prove a theorem for bilinear forms analogous to Theorem 3. 55. The Rank of the Adjoint Form, Suppose the discriminant a n of the quadratic form 'LuyXiXj is pf rank r, and that the discrimi- n nant A of its adjoint S A^UfUj is of rank B. Then, iir<n — l, all the (n — l)-rowed determinants of a are zero ; but these are the ele- ments of A, hence R=0. If »• = w — 1, at least one of the elements of A is not zero, and all two-rowed determinants of A are zero (since by § 11 each of them contains « as a factor), hence ^=1. If r= w, Ii = n; for if H were less than n we should have ^ = 0, and there- fore a= (since A = a"~i). But this is impossible, since by hypothesis r = n. We have then : Theoeem 1. If the rank of a quadratic form in n variables and of its adjoint are r and li respectively, then ifr = n, R = n, if r = n—l, i2 = l, ifr<n—l, E=0, u 162 INTRODUCTION TO HIGHER ALGEBRA Let us consider further the case r = n — l. Here we have seen that ^ = 1, that is, that the adjoint is the square of a linear form, n n n Comparing coefficients, we see that All the OS cannot be zero, as otherwise we should have ^ = 0. Let e), 4=- 0. Then since ^ _ ^2 j, q we see that not all the quantities {^A^y, ••■ A^„) are zero. Accord- ingly (cf. §44) the point (A^-^, A^^, ■•■ A^„), and therefore also the point ((?j, • • ■ 6'„), is a vertex of the original quadratic form. Thus we have the theorem : Theorem 2. If the rank of a quadratic form in n variables is n—1, its adjoint is the square of a linear form, and the coefficients of this linear form are the coordinates of a vertex of the original form. Since, in the case we are considering, all the vertices of the quadratic form are linearly dependent on any one, this theorem com- pletely determines the linear form in question except for a constant factor. CHAPTER XIII PAIRS OF QUADRATIC FORMS 56. Pairs of Conies. We will give in this section a short geomet- rical introduction to the study of pairs of quadratic forms, confining ourselves, for the sake of brevity, to two dimensions. Let u and v be two conies which we will assume to be so situated that they intersect in four, and only four, distinct points, A, B, C, D. Consider all conies through these four points. These conies, we will say, form a pencil. It is obvious that there are three and only three singular conies (i.e. conies which consist of pairs of lines) in this pencil, namely, the three pairs of lines AB, CD ; BC, DA ; AG, BD Let us call the " vertices " of these conies P, Q, and R respectively. From the harmonic properties of the complete quadrilateral* we see that the secants PAB and POD are divided harmonically by the • Cf . any book on modern geometry. 163 J64 INTRODUCTION TO HIGHER ALGEBRA line QR. Accordingly QR is the polar of P with regard to every conic of the pencil. In a similar manner PR is the polar of Q, and PQ the polar of R with regard to every conic of the pencil. Thus, we see that the triangle PQR is a self-conjugate triangle (see §41) with regard to every conic of the pencil. Accordingly, if we per- form a coUineation which carries over P, Q, R into the origin and the points at infinity on the axes of x and y, the equation of every conic of the pencil will he reduced to a form in which only the square terms enter. We are thus led to the result : t Theorem, if two conies intersect in four and only four distinct points, there exists a non-singular coUineation which reduces their equations to the normal form f Aj^x^ + A^xl + A^l = 0, 1 B^xl + B^xl + B^xl = 0. If we wish to carry through this reduction analytically, we shall write the equations of the two conies u and v in the forms (1) taif>iXj = 0, ^b^XiXj = 0. The pencil of conies may then be written 3 (2) SK--X5i,>ia;,.= U, 1 or rather, to be accurate, this equation will represent for different values of X all the conies of the pencil except the conic v. The singular conies of the pencil will be obtained by equating the discriminant of (2) to zero. (3) aji-XJii «12 - ^^^12 ^13 ~ ^"13 «21 - ^^21 «22 - ^*22 «23 - ^*23 agi - XJgi «32 - ^^^32 agg— XOgj = 0. This equation we will call the \-equation of the two conies When expanded, it takes the form (4) - A'X8 -1- @'\= - ® X -1- A = 0, PAIRS OF QUADRATIC FORMS 165 where A, A' are the discriminants of u and v respectively, and = hi *31 *12 hi *32 "83 + *11 hi *31 "12 ^22 "32 *33 + "ll hi "31 •«12 *22 "62 *38 while 0' can be obtained from ® by an interchange of the letters a and b. It can readily be proved (cf. the next section) that the co- efficients @ and & as well as A and A' are invariants of weight two. Except when the discriminant A' of v is zero, the equation (4) is of the third degree, and its three roots, which in the case we have considered must evidently be distinct, give, when substituted in (2), the three singular conies of the pencil. We will not stop here to show how the theory of any two conies, where no restriction as to the number of points of intersec- tion is made, can be deduced from equation (3).* This will follow in Chapter XXII as an application of the method of elementary divisors. Our only object in this section has been to give a geo- metrical basis for the appreciation of the following sections. 57. Invariants of a Pair of Quadratic Forms. Their X-Equation. We consider the pair of quadratic forms n n and form from them the pencil of quadratic forms n The discriminant of this pencil, a-, *ii ~ ''^^11 *i« — "^hn \i ~~ ^Ki ,. — ^.5„ =^(X), is a polynomial in X which is in general of degree n, and which may be written ^(x) = ©„ - ©jX + • • • + ( - 1)»@„X«- * An elementary discussion of the X-equation of two oonios (Vequation en \) is regularly given in French text-books on a;nalytic geometry. See, for instance, Briot et Bouquet, Leqons de GeomHrie analytique, 14th ed., p. 349, or Niewenglowski, Oours de Qeometrie analytique, "Vol. I, p. 459. 166 INTRODUCTION TO HIGHER ALGEBRA The coefficients of this polynomial are themselves polynomials in the a^-'s and J.^'s, ©o and 0„ being merely the discriminants of <j> and ■\jr respectively, while 0^ is the suni of all the different determinants which can be formed by replacing k columns of the discriminant of (f) by the corresponding columns of the discriminant of yjr. Theorem 1. The coefficients @q, ■■■®„of F{'X.) are integral rational invariants of weight two of the pair of quadratic forms (f>, i^.* In order to prove this, let us consider a linear transformation of determinant c which carries over <^ and ■yjr into ^' and ■^' respec- tively, where » ^' = '2a^jx'iX^, Let us denote by ©J the polynomial in the a^'s and b'/s obtained by putting accents to the a's and 6's in @j. Our theorem will then be proved if we can establish the identities &i = c^@i (i=0,l,-n) This follows at once from the fact that F{\), being the discriminant oi <j> — \i/p, is an invariant of weight two, so that if we denote by J"(\) the discriminant of (j)' — \i/r', we have F'(X) = c2I'(\). This being an identity in X as well as in the a's and J's, we can equate the coefficients of like powers of \ on the two sides, and this gives precisely the identities we wished to establish. | The equation F(x\ = we will call the X-equation of the pair of forms </>, yjr. Since, as we have seen, F is merely multiplied by a constant different from zero when ^ and y]r are subjected to a non-singular linear transformation, * Cf. Exercise 13, § 90. f The method by which we have here arrived at invariants of the system of two quadratic forms will be seen to be of very general application. If we have an integral rational invariant /of weight ^ of a single form of the kth degree in ?i variables, we can find a large number of invariants of the system <t>i,<t>2, •■•(pp oip forms of the feth degree in n variables by forming the invariant /for the form Xi0i + ... +Xp0p. This will be a polymonial in the X's, each of whose coefBcients is seen, precisely as above, to be an integral rational invariant of the systems of ^'s of weight /i. PAIRS OF QUADRATIC FORMS 167 the roots of the X-equation will not be changed by such a transfor- mation. These roots, however, are irrational functions of the 0's and hence of the a's and 6's. We may therefore state the result : Theorem 2. The roots of the X-equation of a pair of quadratie forms are absolute irrational invariants of this pair of forms with regard to non-singular linear transformations. It is clear that the multiplicity of any root of the \-equation will not be changed by a non-singular linear transformation. Hence Theorem 3. The multiplicities of the roots of the \-equation are arithmetical invariants of the pair of quadratic forms with regard to non-singular linear transformations- If 4> = a^x\+ ■■■ + a„x% ■\fr= a;f + ••• -^ a;2, the roots of the X-equation are a^, ■■• a„. This example shows that the absolute invariants of Theorem 2 may have any values, and also that the arithmetical invariants of Theorem 3 are subject to no other restriction than the obvious one of being positive in- tegers whose sum is n. 58. Reduction to Normal Form when the \-Equation has no Multi- ple Roots. Although our main concern in this section is with the case in which the \-equation has no multiple roots, we begin by estab- lishing a theorem which applies to a much more general case. Theorem 1. If Xj is a simple root of the \-equation of the pair <f>, ■^{r of quadratic forms in n variables, then, by a non-singular linear transformation, (f> and yjr can be reduced respectively to the forms where e^ is a constant not zero and ^j, i/rj are quadratic formB in tht n— 1 variables z^, ■■• z„. To prove this, we will consider the pencil of forms (2) ^-Xf = <^-Xjf + (Xi-X)'f. 168 INTRODUCTION TO HIGHER ALGEBRA Since \j is a root of the X.-equatioa of the pair of forms <f), yjr, the form ^ — Xji/r is singular, and can "therefore, by a suitable non-singular linear transformation, be written in a form in which one of the variables, say x[, does not enter, ^ _ \j^ = (f>i(x'^, . . . z'^). If this transformation reduces ijr to yfr', we have (3) <j>-\y{r = <^'(4, •••<) + (\i - X)f'{x[, ■ ■ ■ a/„). The discriminant of the quadratic form which stands here on the right cannot contain \j — \ as a factor more than once, since \j is, by hypothesis, not a multiple root of the X-equation of <f> and yjr. It follows from this that the coefficient of x!^ in the quadratic form yjr' cannot be zero, for otherwise the discriminant of the right-hand side of (3) would have a zero in the upper left-hand corner, and Xj — X, would be a factor of all the elements of its first row and also of its first column ; so that it would contain the factor (Xj — X)^. Since the coefficient of x'^ in i/r' is not zero, we can by Lagrange's reduction (Formulte (2), (3), §45) obtain a non-singular linear trans- formation of the form ■ Si = 7ia;i + 72^ + - + Jn^'n Z^ = which reduces ■^' to the form This transformation carries over the second member of (3) into <l>'(z^, ••■3„)+(Xi-X)^i(22, ••■z„)-J-(Xi-X)ci4 Combining these two linear transformations and writing f (^2' ••• 2») + ^^1^1(22' ■•■^n) = <f>l{h^ ••• ^n), we have thus obtained a non-singular linear transformation which effects the reduction. If, here, we equate the coefficients of X on both sides, and the terms independent of X, we see that we have precisely the reduc- tion of the forms ^, yjr to the forms (1); and the theorem is proved. PAIRS OF QUADRATIC FORMS 169 Let us now assume that tlie form ■yfr is non-singular, thus insur- ing that the \-equation be of degree n. We will further assume that the roots Xj, \, •■■ X„ of this equation are all distinct. We can then, by the theorem just proved, reduce the forms <f>, ^fr to the forms (1) by a non-singular linear transformation. The \-equation of these two forms is seen to differ from the X-equation of the pair of forms in {n — 1) variables ^j, ■^-^ only by the presence of the extra factor \j — \. Accordingly the \-equation of the pair of forms (^j, i^j has as its roots Xji •" ^n ^"^^ these are all simple roots. We may there- fore apply the reduction of Theorem 1 to the two forms </)j, -i/tj and thus by a non-singular linear transformation of z^ ■■■ s„ reduce them to the forms \c^z'^ -F ^^{z\, ■■■ sj,). This linear transformation may, by means of the additional formula be regarded as a non-singular linear transformation of Sj, ••• s„ which carries over ^, ■^ into the forms Cisi2+ ^242+^2(4, ...2;). Proceeding in this way, we establish the theorem : Theorem 2. If 0, i|r are quadratic forms in (x-^, ■ ■ ■ x„) of which the second is non-singular, and if the roots \, ■■■ X„ of their \-equation are all distinct, there exists a non- singular linear transformation which carries over and -f into ^^ ^^^12 ^x^c^x'^^ + - + X„c„<2, c^x[^+ c^x'i-h ■■■ + c„42 respectively, where Cj,--- c„ are constants all different from zero. Since none of the c's are zero, the linear transformation x'!=^ix'i (i=l, 2, -m) is non-singular. Performing this transformation, we get the further result : Theorem 3. Under the same conditions as in Theorem 2, ^ and i|r may be reduced hy means of a non-singular linear transformation to the normal forms \^x\ + \^xl-^ - -^ X„4, a;f-f- a;|-f- ■•• -|- xl 170 INTRODUCTION TO HIGHER ALGEBRA From this we infer at once Theorem 4. If in the two pairs of quadratic forms <^, yfr and <j)', T|r' the forms -yjr and ijr' are both non singular, and if the \-equations of these two pairs of forms have no multiple roots, a necessary and suf- ficient condition for the equivalence of the two pairs of forms is that these two X-equations have the same roots ; or, what amounts to the same thing, that the invariants @q, @j, • ■ • ©„ of the first pair of forms he pro- portional to the invariants ©q, ®[, • • • ©» of the second. EXERCISE Prove that, under the conditions of Theorem 3, the reduction to the normal form can be performed in essentially only one way, the only possible variation consisting in a change of sign of some of the x's in the normal form. 59. Reduction to Normal Form when v|f is Definite and Non- singular. We now consider the case of two real quadratic forms <^, i/r of which i|r is definite and non-singular. Our main problem is to reduce this pair of forms to a normal form by means of a real linear transformation. For this purpose we begin by proving Theorem 1. The X-equation of a pair of real quadratic forms <}>, yjr can have no imaginary root if the form yfr is definite and non- singular. For, if possible, let a + /3i {a and y8 real) be an imaginary root of this \-equation, so that /3 =?i= 0. Then (j> — a-\jr — i/3-\jr will be a singular quadratic form, and can therefore be reduced by a non-singular linear transformation ^n = {Pm + iq,a)Xl+ •■■+ (j>nn + iqnn)Xn to the sum of k squares, where k < n, (1) 4>-a^}r-i|3y|r = x[^-{-x!^+ -+X/,?. Let (2) !/l=PllXi+ ■■■+PlnXn, (3) Si = qiiXi + •■■ +qinX„ so that Xi = yi+ iz, . "«> PAIRS OF QUADRATIC FORMS 171 By equating the coefficients of i on the two sides of (1) we thus get (4) -^^ = 2yi2i + 2y2'22+ -+2^,34. Let us now determine x^, ■■■ x„ so as to make the right-hand side of (4) vanish, for instance by means of the equations A reference to (2) shows that we have here a system of k real homogeneous linear equations in n unknowns, so that real values of a;j, ••• a;„not all zero can be found satisfying these equations. For these values of the variables, we see from (4) that yjr vanishes ; but this is impossible (cf . the Corollary of Theorem 3, § 52), since yjr is by hypothesis non-singular and definite. Theorem 2. If yjr is a non-singular definite quadratic form and <f) is any real quadratic form^ the pair of forms (f>, i|r can be reduced by a real non-singular linear transformation to the normal form (5) (j>=±(\,x'^-h-+Kx'i), ylr=±{ x[^+-+ x'i). where Xj, • • • \„ are the roots of the X-equation, and the upper or lower sign is to be used in both cases according as ^jr is a positive or a negative form. The proof of this theorem is very similar to the proof of Theorem 2, § 58. We must first prove, as in Theorem 1, § 58, that <j), yfr can be reduced by a real non-singular linear transforma- tion to the forms , , • s ^ [ C^zf-f l/rj(22, ■■•3„). To prove this, we consider the, pencil of forms ^-\f = ^- X^f -^ (Xj - X)yjr. Since \ is real by Theorem 1, — Xji/r is a real singular quad- ratic form, and can therefore by a real non-singular linear trans- formation be reduced to a form in which one of the variables does not enter, , . , ,,, , ,. 172 INTRODUCTION TO HIGHER ALGEBRA If this transformation reduces ■yjr to i/r', we have (7) <|,-Xf = c|>'{4,■■■xi) + {X,-\)^lr'{x'„■..xl). At this point comes the essential difference between the case we are now considering and the case considered in § 58, as Xj may now be a multiple root of the discriminant of the right-hand side of (7). We need, then, a different method for showing that the coefficient of x'^ in i/r' is not zero. For this purpose it is sufficient to notice that y]r, and therefore also yfr', is a non-singular definite form, and that accordingly, by Theorem 4, § 52, the coefficient of none of the square terms in ■^' can be zero. Having thus shown that the coefficient of x'-^ in yfr' is not zero, we can apply Lagrange's reduction to tlr', and thus complete the reduction of the forms (f>, yfr to the forms (6) precisely as in the proof of Theorem 1, § 58, noticing that the transformation we have to deal with is real. In (6), <^j, -v/rj are real quadratic forms in the n — 1 variables Sg, ■•• 3„. Moreover, since is non-singular and definite, it follows that the same is true of yjr^. For, if -Jcj were either singular or indefinite, we could find values of z^, ■■■ z„ not all zero and such that i^j = ; and these values to- gether with the value Sj = would make -v|r = 0. This, however, is impossible by the Corollary of Theorem 3, § 52. The \-equation of the two forms ^j, yjr-^ evidently differs from the X-equation of <^, yfr only by the absence of the factor \ — \j. The roots of the \-equation of <^j, yjr.^ are therefore \, ■•■ X„, so that if we reduce <f>-^ and t/tj by the method already used for <f>, yfr (we have just seen that <f>■^^, i/tj satisfy all the conditions imposed on cj), yfr), we get ^i(«2' ••• Zn) = \o^4^+ ^2(4' •■• 2»> ti(32, • • • 2!„) = <?222^ + ^2(4' " ' " O' Proceeding in this way, we finally reduce ^,yfrhy a, real non-singu- lar linear transformation to the forms \yfr= c^y\+---+c„yl. PAIRS OF QUADRATIC FORMS 173 Since yjr is definite, the constants Cj, • ■ • c„ are all positive or all nega- tive according as i/r is a positive or a negative form. By means of the further non-singular real linear transformation a'i=V[^|yi (i=l, 2, ..-m), the forms/(8) may be reduced to the forms (5), and our theorem is proved. EXERCISES 1. If (^ is a real quadratic form in n variables of rank r, prove that it can be reduced by a real orthogonal transformation in n variables to the form Cf . Exercises, § 52. 2. Show that the determinant of the orthogonal transformation of Exercise 1 may be taken at pleasure as + 1 or — 1. 3. Discuss the metrical classification of real quadric surfaces along the following lines: Assume the equation in non-homogeneous rectangular coordinates, and show that by a transformation to another system of rectangular coordinates having the same origin the equation can be reduced to a form where the terms of the second degree have one or the other of the five forms (the A'a being positive constants) ■Aixl + A2XI+ AsXs, A i^i + A^xl — AgXg, Aixl + Aix\, Aixl. Then simplify each of the non-homogeneous equations thus obtained by further transformations of coordinates; thus getting finally the standard forms of the equations of ellipsoids, hyperboloids, paraboloids, cones, cylinders, and planes ■which are discussed in all elementary text-books of solid analytic geometry. CHAPTER XIV SOME PROPERTIES OF POLYNOMIALS IN GENERAL 60. Factors and Reducibility. In the present section we will introduce certain conceptions of fundamental importance in our subsequent work. Definition 1. By a factor or divisor of a polynomial f in any number of variables is understood a polynomial (f> which satisfies an identity of the form j.- = ±^ iff being also a polynomial. It will be noticed that every constant different from zero is a factor of every polynomial ; that every polynomial is a factor of a polynomial which vanishes identically; while a polynomial which is a mere constant, different from zero, has no factors other than constants. We note also that a polynomial in x-^, ■■• x„ which is not identically zero cannot have as a factor a polynomial which actually contains any other variables. The conception of reducibility, which we have already met in a special case (§47), we define as follows: Definition 2. A polynomial is said to he reducible if it is iden- dcally equal to the product of two polynomials neither of which is a constant. > In dealing with real polynomials, a narrower determination of the conception of reducibility is usually desirable. We consider, then, what we will call reducibility in the domain of reals, a con- ception which we define as follows: Definition 3. A real polynomial is said to he reducible in the domain of reals if it is identically equal to the product of two othef real polynomials neither of which is a constant. m SOME PROPERTIES OF POLYNOMIALS IN GENERAL 175 In many branches of algebra still another modification of the conception of reducibility plays an important part. In order to explain this, we first lay down the following definition : DBFiUfiTiON 4. A set of numbers is said to form a domain of rationality if, when a and b are any numbers of the set, a+ b, a~b, db, and, so far as b^O, a/b are also numbers of the set. Thus all numbers, real and imaginary, form a domain of ration- ality, and the same is true of all real numbers. The simplest of all domains of rationality, apart from the one which contains only the single number zero, is what is known as the natural domain, that is all rational numbers, positive and negative. A more complicated domain of rationality would be the one consisting of all numbers of the form a + b V — 1, where a and b are not merely real, but rational. These illustrations, which might be multiplied indefinitely, should suffice to make the scope of the above definition clear.* Definition 5. A polynomial all of whose coefficients lie in a domain of rationality M is said to be reducible in this domain if it is identically equal to a product of two polynomials, neither of which is a constant, whose coefficients also lie in this domain. It will be noticed that Definitions 2 and 3 are merely the special cases of this definition in which the domain of rationality is the domain of all numbers, and the domain of all reals respectively. To illustrate these three definitions, we note that the polynomial x^ + 1 is reducible according to Definition 2, since it is identically equal to (a; + V — 1) (a; — V — 1). It is, however, not reducible in the domain of reals, nor in the natural domain. On the other hand, x^ — 2 is reducible in the domain of reals, but not in the natural domain. Finally, ai^ _ 4 is reducible in the natural domain. Leaving these modifications of the conception of reducibility, we close this section with the following two definitions : Definition 6. Two polynomials are said to be relatively prime ij they have no common factor other than a constant. *ByiJ (01,02, ••■an) is understood the domain of rationality consisting of all numbers which can be obtained from the given numbers ai, ■•• a» by the rational pro- cesses (addition, subtraction, multiplication, and division). In this notation the natural domain vfould be most simply denoted by 72(1); the domain last mentioned in the text by B (1, V^l) or, even more simply, by B {V^l). This notation would not apply to all cases {_e.g. the real domain) except by the use of an infinite number of arguments. 176 INTRODUCTION TO HIGHER ALGEBRA Definition 7. Two methods of factoring a polynomial shall be said to he not essentially different if there are the same number n of factors in each case, and these factors can he so arranged that the kth factors are proportional for all values of k, from 1 to n inclusive.. EXERCISES 1. Prove that every polynomial in (x, y) is irreducible if it is of the form /(^) + y, where / (x) is a polynomial in x alone. Would this also be true for poljmomials of the form 2. li f,<j>,}p are polynomials in any number of variables which satisfy the relation f=4>^, and if the coefficients of /and <f> lie in a certain domain of rationality, prove that the coefficients of i/» will lie in the same domain provided <^ ^ 0. , 61. The Irreducibility of the General Determinant and of the Sym- metrical Determinant. Theorem 1. The determinant D = ni "'12 "■!» ^ix '''22 "■ '"Z" ^n% is an irreducible polynomial if its r? elements are regarded as inde- pendent variables. For suppose it were reducible, and let where neither/ nor ^ is a constant. Expanding D according to the elements of the first row, we see that it is of the first degree in a^j. Hence one of the two polynomials / and <f) must be of the first degree in ajj, the other of the zeroth degree. Precisely the same reasoning shows that if a,;,- is any element of Z>, one of the polyno- mials / and (j) will be of the first degree in a^p while the other will not involve this variable. Let us denote by/ that one of the two polynomials which involves %, any element of the principal diagonal of D. Then <f> does not SOME PROPERTIES OF POLYNOMIALS IN GENERAL 177 involve any element of the ith row or the ith column. For if it did, since / is of the first degree in a^ and <f) is of the zeroth, their product D would involve terms containing products of the form %a^y or a^aji, which, from the definition of a determinant, is impossible. Consequently, if either one of the polynomials / and (j) contains any element of the principal diagonal of D, it must contain all the ele- ments standing in the same row and all those standing in the same column with this one, and none of these can occur in the other polynomial. Now suppose/ contains a« and that ^ contains any other element of the principal diagonal, say a^j. Then «;,- and a,-; can be in neither/ nor (j>, which is impossible. Hence, if / contains any one of the ele- ments in the principal diagonal, it must contain all the others, and hence all the elements, and our theorem is proved. Theorem 2. The symmetrical determinant i) = "In *nl (S- — "'Ji) is an irreducible polynomial if its ^n(n + l) elements be regarded as independent variables. The proof given for the last theorem holds, almost word for word, in this case also, the only difference being that while D is of the first degree in each of the elements of its principal diagonal, it is of the second degree in each of the other elements. The slight changes in the proof made necessary by this difference are left to the reader. EXERCISES 1. The general bordered determinant an- ■ ain «n • ■ «ij, a„i . • «nm «nl- .M«p Vn. • Vi„ 0. .0 V 0...0 is irreducible iip<n, the a's, u's, and «'s being regarded as independent variables 2. The symmetrical bordered determinant obtained from the determinant is Exercise 1 by letting a<, = aj,, u,^ = Vji is irreducible iip<n. 178 INTRODUCTION TO HIGHER ALGEBRA 3. If for certain values of i and J, but not for all, a^f = Ojt, but if the o's are otherwise independent, can we still say that an . ■ . Qin is irreducible ? Onl . 4. Prove that a skew-symmetric determinant (of. Exercises 2, 3, §20) is always reducible by showing that, when it is of even order, it is a perfect square. [Suggestion. Use Corollary 3, § 11, and Theorem 6 and Exercise 1, § 76.] Does this theorem require any modification if the elements are real and we consider reduoibility in the domain of reals ? 62. Corresponding Homogeneous and Non-Homogeneous Polyno- mials. It is often convenient to consider side by side two polyno- mials, one homogeneous and the other non-homogeneous, which bear to one another the same relation as the first members of the equations of a plane curve or of a surface in homogeneous and non- homogeneous coordinates respectively. Such polynomials we will speak of as corresponding/ to one another according to the following definition : Definition. If we have a non-homogeneous polynomial of the Jcth degree in any number of variables (ajj, ••■2'n_i) and form a new poly- nomial by multiplying each term of the old by the power of a new variable a-„ necessary to bring up the degree of this term to k, the homo- geneous polynomial thus formed shall be said to correspond to the given non-homogeneous polynomial. Thus the two polynomials (1) 2a^ + 3x^y-5xz^-i/2 +2z^ +x -3y -9, (2) 2x^-\- Sx^y - 5xz^ - yzt + 2zH + xt^-2,yt^ - 9«8, correspond to each other. It may be noticed that if <^ (a;j, '"Xn-^ is the non-homogeneous polynomial of degree A, the corresponding homogeneous polynomial may be written , /a- a- g- \ To every non-homogeneous polynomial there corresponds one, and only one, homogeneous polynomial. Conversely, however, to a homogeneous polynomial in n variables there correspond in general SOME PROPERTIES OF POLYNOMIALS IN GENERAL 179 n different non-homogeneous polynomials which are obtained by set- ting one of the variables equal to 1. For instance, in the example given above, to (2) corresponds not only (1) but also (3) 2 +32^ -5z^ -yzt+2eH + t^ -Syt^-9i?, (4) 2a^ + 3a;2 -5xz^-zt +2zH + xfi-Bt^ -9f, (5) 23? + Sx^i/-5x -yt +2t +xt^-Syt^ -9^. It should be noticed, however, that if one of the variables enters into every term of a homogeneous polynomial, the result of setting this variable equal to "unity is to give, not a corresponding non-homo-* geneous polynomial, but a polynomial of lower degree. In fact, in the extreme case in which every variable enters into every term of the homogeneous polynomial, there is no corresponding non-homo- geneous polynomial; as, for instance, in the case of the polynomial x^yz + xyh, -t- xy^. Theorem 1. If one of two corresponding polynomials is redueihle^ then the other is, also, and the factors of each polynomial correspond to the factors of the other. For let <]){x^, ■ ■ ■ Xn) be a homogeneous polynomial of degree (k + 1), and suppose it can be factored into two factors of degrees k and I, respectively, Now suppose the corresponding non-homogeneous polynomial in question is the one formed by setting x^ = 1- We have Since by hypothesis the degree of the polynomial on the left is unchanged by this operation, neither of the factors on the right- hand side of (6) can have had its degree reduced, hence neither of the factors on the right of (7) is a constant. Our non-homo- geneous polynomial is therefore reducible ; and moreover the two factors on the right of (7), being of degrees k and I respectively, are precisely the two functions corresponding to the two factors on the right of (6). 180 INTRODUCTION TO HIGHER ALGEBRA Now, let ^i+i(a;i, ■•• Xn-y) be a non-homogeneous poljmomial, and ^*+i(*i> • ■ • a^n-i) = %(aii • • • a;^i) Xi(a;i, • • ■ a;^i), where the subscripts denote the degrees of the polynomials. Let 4>k+i^ "^k+ii Xk+i be the homogeneous polynomials corresponding to *, ■*•, X. Then when x„ ^ 0, * h+l \x„ x„ J \^„ x„ J \x„ x„ J .Multiplying this equation by a;^"*"' we have an equation which holds whenever x^ =#= 0, and, therefore, by Theorem 5, § 2, is an identity. Thus our theorem is proved. As a simple illustration of the way in which this theorem may be applied we mention the condition for reducibility of a non-homo- geneous quadratic polynomial in any number of variables. By applying the test of § 47 to the corresponding homogeneous poly- nomial we obtain at once a test for the reducibility of any non- homogeneous quadratic polynomial. Theokbm 2. If f and ^ are non-homogeneous polynomials, and F, <I> are the corresponding homogeneous polynomials, a necessary and sufficient condition that F and <1> he relatively prime is that f and (j> be relatively prime. For if/ and ^ have a common factor i/r which is not a constant, the homogeneous polynomial '^ which corresponds to i|r is, by Theorem 1, a common factor of F and $, and is clearly not a con- stant. Conversely, if 'Sf is a common factor of F and <I> which is not a constant, / and </> will have, by Theorem 1, a common factor which corresponds to ^ and which therefore cannot be a mere constant. 63. Division of Polynomials. We will consider first two polyno- mials in one variable: '»> (X) {f{x)~a^3f' + a^x^-^+... + a, \4>(x) = h^x'^ + J^a;"-! -(-■•■ -f 6„. SOME PROPERTIES OF POLYNOMIALS IN GENERAL 181 We learn in elementary algebra how to divide/ by 0, getting a quotient Q (x) and a remainder R (x). What is essential here is contained in the following theorem : Theorem 1. If f and <j} are two polynomials in x of which is not identically zero, there exists one, and only one, pair of polynomials, Q and S, which satisfy the identity (2) f{x)=Q{x)cj>{x) + E{x), and such that either B = Q,* or the degree of M is less than the degree ofcj). We begin by proving that at least one pair of polynomials Q, Jt exists which satisfies the conditions of the theorem. If /is of lower degree than <j) (or if /= 0), the truth of this state- ment is obvious, for we may then let Q = 0, R =/. Suppose, then, that/ is of at least as high degree as ^. Writing / and (ji in the form (1), we may assume asg gb 0, Jfl =?t 0, n> m. Lemma, if cf *'* '^^^ ^f higher degree thanf, there exist two polyno- mials Qi and R^ which satisfy the identity f{x)=Q,(x)<j>{x) + R,ix), and such that either Ri=0, or the degree of R^ is less than the degree off. The truth of this lemma is obvious if we let These two polynomials Q-^ and R^ will serve as the polynomials Q and R of our theorem if R^=0, or if the degree of R^ is less than the degree of <^. If not, apply the lemma again to the two functions R^ and 0, getting j^^ (-^^ ^ q^ (^^ ^ (■^^ + r^ (^), where R^ is either identically zero or is of lower degree than R^ We may then write, y(^) = ^ q^ (^^ + q^ (^)] ^ (a;) + R^ (x). If R^ = 0, or if the degree ofR^ is less than the degree of ^, we may take for the polynomials Q, R of our theorem, Q^ + Q^ and R^. If not, we apply our lemma again to R^ and (p. Proceeding in this way * It will be remembered that, according to the definition we have adopted, a polyno- mial which vanishes identically has no degree. 182 INTRODUCTION TO HIGHER ALGEBRA we get a series of polynomials Mi, R^, ■■■ whose degrees are con- stantly decreasing. We therefore, after a certain number of steps, reach a polynomial Ri which is either identically zero or of degree less than ^. Combining the identities obtained up to this point, we I'^^e f{x)^\_Qi{x)+.:+Q, (x) ] (^) + R, {x), an identity which proves the part of our theorem which states that at least one pair of polynomials Q, R of the kind described exists.* Suppose now that besides the polynomials Q, R of the theorem there existed a second pair of polynomials Q', R' satisfying the same conditions. Subtracting from (2) the similar identity involving Q', R\ we have (3) Q = {Q-Q')<I> + {R-R'). From this we infer, as was to be proved, Q=Q\ R = R'. For if Q and Q' were not identical, the first term on the right ol (3) would be of at least the mth degree, while the second involves no ■power of X as high as m. Turning now to polynomials in several variables : 1 <¥,Xv ■ ■ ■ ^k) = *o(^2' • ■ • ^^^"\ + *l(2'2' ■ • • «*>r"^ + • • • + K{^V ■ ■ ■ -Vk), the ordinary method of dividing/ by </> would give us as quotient and remainder, not polynomials, but fractional rational functions. In order to avoid this, we state our theorem in the following form : Theokem 2. Iff and (j) are polynomials in {x-^, ■■■ x^) of which is ^'ot id! entically zero, there exist polynomials Q, R, _P, of which the last is not identically zero and does not involve the variable a;-^, which satisfy the identity, (5) P(a-2, ••• x,)f{xi, ■■■ x^) = Q{xi, ■■■ Xi)(j){xi, ••• X;,) + R{xi, ••• x^), and such that either R = 0,or the degree in x^ of R is less than the degree in x^ of (p. The proof of this theorem follows the same lines as the proof of Theorem 1. * The reader should notice that the process just used Is merely the ordinary process of long division. SOME PROPERTIES OF POLYNOMIALS IN GENERAL 183 If/ is of lower degree in x-^ than <^ (or if/= 0), the truth of the theorem is obvious, for we may then let P= 1, ^ = 0, R=f. Suppose, then, that / is of at least as high degree in x^ as ^. "Writing /and <^ in the form (4), we may assume flSj ^ 0, Jfl ^ 0, n> m. Lemma. If <j) is not of higher degree in x^ than /, there exist two polynomials Q-y, M^ which satisfy the identity, and such that either R^ = 0, or the degree of R^ in x-^ is less than the degree of fin Xy The truth of this lemma is obvious if we let ^1 — "^oC^a' ■ ■ ■ ^*) *: "" • The polynomials Q^ R-^, h^ will serve as the polynomials Q, R, P of our theorem if R^ = 0, or if the degree of R^ in a^j is less than the degree of ^ in a;j. If not, apply the lemma again to the two functions Ry and <^, getting where R^ is either identically zero or is of lower degree in x-^ than JBj. We may then write 52/^(5^^^+ Q^)^ + R^. If R^ = 0, or if the degree of R^ in x^ is less than the degree of (/> in Xy we may take for the polynomials Q, R, P of our theorem the functions Jo^i+ Qv -^2' ^o* ^^ ''^^^■> ^® ^PP^y our lemma again to R^ and <^. Proceeding in this way, we get a series of polynomials Ry i^g, •••whose degrees in x^ are constantly decreasing. We there- fore, after a certain number of steps, reach a polynomial R^ which is either identically zero, or of degree in x^ less than <^. Combining the identities obtained up to this point, we have hif=(hl-^Q^ + hl^Q^+.: + Q,)<i, + R,, an identity which proves our theorem, and which also establishes the additional result : Corollary. The polynomial P whose existence is stated in out theorem may he taken as a power of 6q. 184 mTRODUCTION TO HIGHER ALGEBRA We note that it would obviously not be correct to add to the statement of Theorem 2 the further statement that there is only one set of polynomials Q, B, P, since the identity (5) may be multi- plied by any polynomial in {x^, •••a;„) without changing its form. Cf., however, the exercise at the end of § 73. 64. A Special Transformation of a Polynomial. Suppose that j\x^,x^,x^,x^) is a homogeneous polynomial of the kth degree in the homogeneous coordinates {x^, x^, x^, x^), so that the equation / = represents a surface of the kth degree. If, in /, the term in a^ has the coef3ficient zero, the surface passes through the origin; and if the term in x^ (or x^, or x^) has the coefficient zero, the surface passes through the point at infinity on the axis of x^ (or x^, or Xg). It is clear that these peculiarities of the surface can be avoided, and that, too, in an infinite variety of ways, by sub- jecting the surface to a non-singular coUineation which carries over any four non-complanar points, no one of which lies on the surface, into the origin and the three points at infinity on the coordinate axes. It is this fact, generalized to the case of n variables, whic'" we now proceed to prove. Lemma. If f{x-^, ••■a;„) is a homogeneous polynomial of the kth degree in which the term 24 is wanting, there exists a non-singular linear transformation of the variables {x-y, ■■■ x„) which carries f into a new form /j, in which the term, in a;,', hag a coefficient different from zero, while the coefficients of the kth powers of the other variables have not been changed. In proving this theorem there is obviously no real loss of gener- ality in taking as the variable x^ the last of the variables x^. Let us then consider the non-singular transformation X, =x[+ afcl (i = 1, •• • » — 1). This transformation carries /into and evidently does not change the coefficients of the terms in ^v --- '* •■xz. SOME PROPERTIES OF POLYNOMIALS IN GENERAL 185 Now, since every term in/j, except the term in x'^, contains at least one of the variables x[,--- x'„^-^, the coefficient of the term in x'" will be /i(0,...,0,l) = /K,-a„_i,l). Our lemma will therefore be proved if we can show that the constants «!, ••• a„_i can be so chosen that this quantity is not zero. Let us take any point (b^, •■■b„) for which &„=?t=0 ; and consider a neighborhood of this point sufficiently small so that a;„ does not vanish at any point in this neighborhood. Then, since / does not vanish identically, we can find a point (c^, ••• c„) in this neighborhood (so that c„#0) such that M, ■••Cn)*0. If now we take for a^, •••«„_! the values c^/c„, •••c„_j/c„, we shall have, since /is homogeneous, Thus our lemma is proved. Theorem 1. If f{x-y,---x^ is a homogeneous polynomial of the kih degree, there exists a non-singular linear transformation which carries k k f into a new form fj^ in which the terms in x\, ■•■ x'„ all have coefficients cHfferent from zero. The proof of this theorem follows at once from the preceding lemma. For we need merely to perform in succession the trans- formations which cause the coefficients first of a;*, then of a;|, etc., to become different from zero, and which our lemma assures us will exist and be non-singular, to obtain the transformation we want. To make sure of this it is necessary merely to notice that the coefficient of 3^ obtained by the first transformation will not be changed by the subsequent transformations ; that the same will be true of the coefficient of a^ obtained by the second trans- formation; etc. Theorem 2. If f{x^, ■■■ x^ is a polynomial of the hth degree which is not necessarily homogeneous, there exists a non-singular homogeneous linear transformation of {x-^, ■■■ x^) which makes this polynomial of the kth degree in each of the variables x\, ••■ x'„ taken separately. 186 INTRODUCTION TO HIGHER ALGEBRA If / is homogeneous, this is equivalent to Theorem 1. If / is non-homogeneous, we may write it in the form where each (^ is a homogeneous polynomial of the degree indicated by its subscript or else is identically zero. We need now merely to apply Theorem 1 to the homogeneous polynomial (jy^, which is, of course, not identically zero. This theorem, and therefore also Theorem 1, which is merely a special case of it, admits the following generalization to the case of a system of functions : Thbokem 3. If we have a system of polynomials of degrees k^, \, ■■■ k^ respectively, there exists a non-singular homo- geneous linear transformation ivhich makes these polynomials of degrees Aj, • • • k^ in each of the variables x[, ■■■ x',^ taken separately. This theorem may be proved either by the same method used ic proving Theorems 1, 2 ; or by applying Theorem 2 to the product /i/a "Vw CHAPTER XV FACTORS AND COMMON FACTORS OF POLYNOMIALS IN ONE VARLABLE AND OF BINARY FORMS 65. Fundamental Theorems on the Factoring of Polynomials in One Variable and of Binary Forms. Theorem 2, § 6 may be stated in the following form : Theorem 1. A polynomial of the nth degree in one variable is always reducible when n>l. It can be resolved into the product of n linear factors in one, and essentially in only one, way. By means of § 62 we can deduce from this a similar theorem in the case of the binary form (1) a^l + ajZ^-'^x^ + ••■ + a„aq. Let us first assume that a^^O. Then the non-homogeneous poly- nomial (2) aQX^ + aj^f-i + ■■■ +a„ corresponds to (1), according to the definition of § 62. Factoring (2), we get «o(^i - «i)(*i - «2) "■ (^1 ~ «»)' or, if we take n constants aj', a'^, ■•• < whose product is «„, (3) «a:i - a[)(a'^Xi - «^) ••• («'X - e/,), where for brevity we have written «'/«. = «J (i=l, 2, ...w). By Theorem 1, § 62, we now infer that the binary form (1) is identi- cally equal to (4) («i'^i - «ia'2)(«2*i - «22^2) ••• («'>i - ««*2)- 187 188 INTKODUCTION TO HIGHER ALGEBRA Moreover, there cannot be any way essentially different from this of factoring (1) into linear factors, for if there were we should, bj setting x^ = 1, have a way of factoring (2) into linear factors essen- tially different from (3). Thus our theorem is proved on the suppo- sition that «Q ^ 0. Turning now to the case a^ = 0, let us suppose that a^= ... = a^_^ = 0, a^^O, where k<n. The form (1) then has the form (5) a^xY'^a* + ■■■ + a^xl, which is equal to the product of k factors x^ and the binary form a^l''' + • ■ ■ + «„a;|-* of degree n — k. Since the first coefficient in this form is not zero, it can, as we have just seen, be factored into n — k linear factors. Thus, here also, we see that the binary form can be written in the form (4), the only peculiarity being that in this case k of the con- stants a" are zero. We leave it to the reader to show that this factoring can be performed in essentially only one way. This being done, we have the result : Theorem 2. A binary form of the nth degree is always reduc- ible when n>\. It can be resolved into the product of n linear factors in one, and essentially only one, way. EXERCISES 1. Prove that every real polynomial in one variable of degree higher than two is reducible in the domain of reals, and can be resolved into irreducible factors in one, and essentially only one, way. 2. Prove the corresponding theorem for real binary forms. 66. The Greatest Common Divisor of Positive Integers.* We will consider in this section the problem of finding the greatest com- mon divisor of two positive integers a and S, which has the closest • In this section we use the term divisor in the arithmetical sense, not in the algebraic sense defined in § 60. An integer h is said to be a divisor of an integer a if an integer c exists such that a = 6c. FACTORS OF POLYNOMIALS IN ONE VARIABLE 189 analogy with the algebraic problem of the next section. The solu- Mon of this problem, which was given by Euclid, is as follows : If we divide a by 6 * and get a quotient g'j and a remainder r^, we may write a = q,b + r^, where, if the division is carried out as far as possible, we. have rj<5. Then divide 6 by r getting a quotient jj and a remainder r^ which, if the division is carried out as far as possible, is less than r^. Proceeding in this way, we get the following system of equations, in which, since the remainders r^, r^, ••• are positive integers which continually decrease, we ultimately come to a point where the divi- sion leaves no remainder : a = qj)+ ri ^^ ^ j^ From the first of these equations we see that every common factor of a and 6 is a factor of r^; from the second, that every common factor of I and r^ is a factor of r^; etc.; finally, that every common factor of rp_2 and r^.-^ is a factor of r^. Hence every common factor of a and b is a factor of r^. On the other hand, we see from the last equation (1) that every factor of j-p is a factor of r^.^ ; from the next to the last equation, that every common factor of j-p and »-p_i is a factor of ri,_^; etc.; finally, that every common factor of r^ and r-^ is a factor of J, and that every common factor of j-j and i is a factor of a. Hence every factor of r^ is a common factor of a and b. Since the largest factor of r,, is r,, itself, we have the result : Theorem 1. In Uuclid's algorithm (1), the greatest common divisor of a and b is r,,. In particular, a necessary and sufficient condition that a and 6 be relatively prime is that r^ = 1. * This is possible even if a < 6, the only peculiarity in this case being that the quotient is zero and the remainder equal to o. 190 INTRODUCTION TO HIGHER ALGEBRA We will next deduce from the equations (1) an important formula by means of which j-p is expressed in terms of a, b, and the g's. From the first equation (1) we have ri = a- q^b. Substituting this value into the second equation, we get for r^ the value ^^ = _ q^a + {q^q^ + V)b. Substituting the values for r^ and r^ just found in the third equation, we get ^^ ^ (^^^^ + i) ^ _ (^^^^^^ + q^ + q,)b. Proceeding in this way, we can express each of the r's, and there- fore ultimately »*p, in terms of a and b. In order to express con- veniently the general formula here, we introduce the following notation : [ j = l [«i] = «i' ^' 1 [«i, ttgi "aJ = "l«2 «3 + «3 + "l' It will be seen that the values of r^, r^, r^ found above are included in the formula (3) »•* = (- 1)*-^ Iqv 1V- ?*-i] « + (-!)* [9o' ?r ?2' • ■ • ?*-!] *• By the method of mathematical induction this formula will therefore be established for all values oi k-^p if, assuming that it holds when k< Aj, we can show that it holds when A = ^j + 1. This follows at once when, in the formula we substitute for r^^, ^^^ ''ai-i their values from (3) and reduce the resulting expression by means of the definitions (2). We have therefore established the formula (4) r, = Aa + Bb, where A = {-iy>-^lq^,q^,...q,_{\, 5 = (- l>'[jo,ji, ... gp_i]. Since the g''s are integers, it is clear that A and B will be integers. FACTORS OF POLYNOMIALS IN ONE VARIABLE 191 The most important application of the result just obtained is to the case in which a and h are relatively prime. Here r^ = 1, and we have (5) Aa + Bb = l. Conversely, if two integers A and B exist which satisfy (6), a and h are relatively prime, as otherwise the left-hand side of (5) would have a factor greater than 1. Theorem 2. A necessary and sufficient condition for a and b to be relatively prime is that there exist two integers A and B such that Aa+Bb = l. EXERCISES 1. Prove that [ai,a2, •■• a^ = [a„,a„-i, •■• ai]. [Suggestion. Use the method of mathematical induction.] 2. Prove that the numerical values of the integers A and B found above are respectively less than ^b and |a. [Suggestion. Show that a/b =[<?oi ■■•??]/[?!) •■• 9p]i ^-nd that this second fraction is expressed in its lowest terms.] 3. Prove that there can exist only one pair of integers A and B satisfying the relation Aa + Bb = \ and such that A and B are numerically less than \ b and J a respectively. 67. The Greatest Common Divisor of Two Polynomials in One Variable. In place of the integers a and b of the last section, we consider here the two polynomials: \f{x)=a^x^-\-a^x'^-^+ •••+«„. 1 </.(:c) = b^x^ + Sja;™-! + . • . + J^. By the greatest common divisor of these two polynomials is meant their common factor of greatest degree.* It will turn out in the course of our work that (except in the case in which/ and ^ are both identically, zero) this greatest common divisor is completely determinate except for an arbitrary constant factor which may be introduced into it. * Many English and American text-books use the term highest common factor ; but as there is not the slightest possibility that the word greatest, here, should refer to the value of the polynomial, since the polynomial has an infinite number of values for dif- ferent values of the argument, it seems better to retain the traditional term. 192 INTRODUCTION TO HIGHER ALGEBRA We will assume that neither/ nor <^ is a mere constant, and that the notation has been so introduced that / is of at least as high degree as ^ ; that is, we assume a„=itO, bo=^0, n>m>0. Let us apply Euclid's algorithm to/ and (f> precisely as in § 66 we applied it to a and b. We thus get the system of identities cl^{x)=Q,{x)R,{x) + R^{x), (2) ^ R^{x)~Qlx)Rlx) + Rlz), [B,.,{x)^Q,{x)RJix) + R,^i. For the sake of uniformity we will write <p{x)^R,(x). Then ^g, i2p R^, ■•• are polynomials of decreasing degrees, so that after a finite number of steps a remainder is reached which is a con- stant. Tliis remainder we have indicated by Rp + i- From this algorithm we infer, as in § 66, that every common factor of /and <^ is a factor of all the R's, and, on the other hand, that every common factor of two successive jB's is a factor of all the preceding K's and therefore of / and (f). Accordingly, if / and ^ have a common factor which is not a constant, this common factor must be a factor of the constant -Kp+j, and therefore 72^^ j = 0. Con- versely, if Rp_^_-^= 0, the polynomial Rp{x) is itself a common factor of Rp and -Bp+j, and therefore of /and (f). Hence the two theorems : Theorem 1. A necessary and sufficient condition that two poly- nomials in one variable f and </>, neither of which is a constant^ be relatively prime is that in Huclid's algorithm, (2), Rp+i # 0. Theorem 2. If in Euclid's algorithm, (2), i?p+ j = 0, then RJ^x) is the greatest common divisor off and ^. By means of this theorem we are in a position to compute the greatest common divisor, not only of two, but of any finite number, of polynomials in one variable. Thus if we want the greatest common divisor of f{x), <f>{x), y}r(x), we should first compute, as above, the greatest common divisor Rp{x) of / and (j), and then, by the same method, compute the greatest common divisor of Iij,(x) and yjr{x). FACTORS OF POLYNOMIALS IN ONE VARIABLE 193 From the identities (2) we can compute the value of each of the remainders in terms of /, (f>, and the quotients Q. The formulae here are precisely like those of § 66, and give for Bp+^ the value (3) B,^^ s ( - ly [ «i {X), Qix), ■ . . Q,(xmx) Suppose, now, that / and <j) are relatively prime. We may then divide (3) by B^+i and get (4) F{x)J{x) + ^{x)4,{x)^l, where (5) ^«-<i^ p+i ( — 1)"+" "P+l Q,{x), Q^x), ... Q^x)^^ From the definitions (2), § 66, we see that F and ^ are polyno- mials in X. The existence of two polynomials F and $ which satisfy (4) is therefore a necessary condition that/ and ^ be relatively prime. It is also a sufficient condition ; for from (4) we see that every com- mon factor of /and <j> must be a factor of 1, that is, must be a con- stant. Thus we have proved the theorem: Theorem 3. A necessary and sufficient condition that the polyno- mials f{x) and 4>(x) ie relatively prime is that two polynomials F{x) and ^{x) exist which satisfy (4).* We can make this statement a little more precise by noting the degrees of F and cE> as given by (5). For this purpose let us first notice that if «j, ■ • ■ <*■„ are polynomials of degrees k-^, ■ • • k„ respec- tively, [«i, • • • «„] will, by (2), § 66, not be of degree greater than /fcj + • • • + /fc„. Now let the degree of Rlx) be %, and, as above, the degrees of /and i^, n and m respectively. Then (cf. (2)) the degrees of ^0' ^1' ^2' ■■■ "^^i^^ be w — m, m — mj, m^—m^,-.. respectively. Ac- cordingly, by (5), the degrees of F and * are respectively not greater than (m — Wj) -I- (% — OTg) ■! |-(wp.i — mp) = m — m,, and {n — m) + {m — m^ + {mi — m^ -\ h(mp_j -Wp) = n — mf,. Hence, since Wp > 0, ^ is of degree less than m, and 4> of degree less than n. * The proof we have given of this theorem applies only vfhen neither/nor tp is a constant. The truth of the theorem is at once obvious if / or is a constant. o 194 INTRODUCTION TO HIGHER ALGEBRA Conversely, we will now show that if F-^ is a polynomial of degree less than m, and ^j a polynomial of degree less than n, and if (6) F^{x)f{x) + <^lx)<^{x) = l, then Ft!^x)^F{x), ^^{x)=^{x). To prove this, subtract (6) from (4), getting {F-F,)f^{_^,-^)j>. If we resolve the two sides of this identity into their linear fac- tors, we see that, since / and ^ are relatively prime, / must be a factor of $j — $ and <^ a factor of F—F-^^. This, however, is pos- sible only if <E>j^— $ and F—F^ vanish identically, as otherwise they would be of lower degree than f and respectively. We have thus proved the theorem : Theorem 4. If f{x) and <f>{x^ are relatively prime, and neither is a constant, there exists one, and only one, pair of polynomials F{x) ^{x), whose degrees are respectively less than the degrees of <^ and f and which satisfy the identity (4). Before proceeding to the general applications of the principles here developed which will be found in the next section, the reader will do well to familiarize himself somewhat with the ideas involved by considering the special case of two polynomials of the second <ieg^ee= f{x) = a^a? + a^x + a^ ag^O, ^(x) = 6q x^ +h^x + h^ h^^^- If the condition that these two polynomials be relatively prime be worked out by a direct application of Euclid's algorithm, it will be found necessary to consider separately the cases in which aj6g — aj)-^ is or is not zero. By collating these results it will be found that in all cases the desired condition is: («2^o - «o*2)^ + («i^o - «o*i)(«/2 - «2*i) ^ 0. This condition may be found more neatly and quickly by obtaining the condition that two polynomials of the form F{x)=p^x+p^, ^{x) = qQX + q^ exist which satisfy the identity (4). It is this last method which we shall apply to the general case in the next section. FACTORS OF POLYNOMIALS IN ONE VARIABLE 195 68. The Resultant of Two Polynomials in One Variable. Let f{x) = a^x" + a^af-'^ + • • • + a„,-^x + a„ a,, :?s= 0, w > 0, ^{x)=h^x^+\x'^-^+ •■■ +h^_^x + h^ Jo^O, m>0. The condition that these polynomials be relatively prime consists as we see from Theorem 4, § 67, in the existence of constants p^ Pv ■■■ Pm-v %" 1v ••■ S'n-i such that (_Poa;"-i+>ia;'»-2+ ... +p^_^)(aaa;' + aiX-^+ - + a„) Equating coefficients of like powers of x, we see that the following system of equations is equivalent to the last written identity: n «oPo +hlo = 0, (Im+lPo + «mi'l+ ••• +«2^».-l +Kqi+ +%m+l = 0, = 0, -1 = 0. -1=0, ^nPm-l + 5m?B-l=l In writing these equations we have assumed for the sake of definiteness that n>m, though the change would be immaterial if this were not the case. This is a system of m + w linear equations in the m + n unknowns jj^, ■ . • Pm-v %■> '" 1n-v whose determinant, after an interchange of rows and columns and a shifting of the rows, is a„ «„ .0 .0 a„ K 6„ .5»o h„. a determinant which, it should be noticed, has m+n rows and columnsk 196 INTRODUCTION TO HIGHER ALGEBRA If -B=?tO, the set of equations (1) has one and only one solution, and / and <j> are relatively prime. If i2 = 0, two cases seem at first sight possible (cf. § 16): either the system of equations has no solu- tion, or it has an infinite number of solutions. This latter alterna- tive cannot, however, really arise, for we have seen in Theorem 4, § 67, that not more than one pair of polynomials F and $ can exist which satisfy formula (4) of that section and whose degrees do not exceed m — 1 and n — l respectively. Aacordingly, if J? = 0, the set of equations has no solution and / and <j) have a common factor. II is called the resultant of /and <f>.* The term resultant has thus been defined only on the supposition that / and ^ are both of at least the first degree. It is desirable to extend this definition to the case in which one or both of these polynomials is a constant. Except in the extreme case m = n=Q, we will continue to use the determinant H as the definition of the resultant. Thus when m = 0, w > we have If 6(1 =#= we have jB ^ 0, and moreover in this case ./ and ^ are relatively prime since the constant ^ has no factors other than constants. If, however, h^ = 0, we have i? = 0, and every factor of /■ is a factor of ^, since ^ is now identically zero. Similarly when w = 0, m > 0, we have and we see that a necessary and sufficient condition that f and <j> be relatively prime is that R^Q. Finally, when w = m = 0, we define the symbol R i "''^\ which we still use to denote the resultant, by the formula "^ 7? (^q\ _ 1 1 when a^, and Jg are not both zero, \hj lO when a^ = 6^ = 0. We may now say with entire generality: Theorem. A necessary and sufficient condition for two polynomials in one variable to be relatively prime is that their resultant do not vanish. For another method of approach to the resultant, cf. Exercise 4 at the end of § 76. * It should be noticed that the resultant of and / may be the negative of the resultant of/ and 4>. This change of sign is of no consequence for most curposes. FACTORS OF POLYNOMIALS IN ONE VARIABLE 197 69. The Greatest Common Divisor in Determinant Form. Definition. By the ith suhresultant iJ; of two polynomials in cne variable is understood the determinant obtained by striking out the first i and the last i rows and also the first i and the last i columns from the resultant of these polynomials. Thus if the polynomials are of degrees 5 and 3 respectively, the resultant B is a determinant of the eighth order, iZj of the sixth, M^ of the fourth, and M^ of the second, as indicated below : R = h «1 «2 «s «4 «5 «6 ^3 «1 «2 «8 «4 «5 «4 *2 ^3 «0 B^= ^0 «1 «2 «3 h h *0 h h f>i h h h K h We now state the following results, leaving their proof to the reader: Lemma. Iffxi^) ^'^'^ </>i(^) *^^ polynomials, and f{x) = {x- a)f,{x), <^{x) = {x- a)ct>,(x), the resultant of f-^ and <f)^ and their successive subresultants are equal respectively to the successive subresultants off and (j). Theoeem 1. The degree of the greatest common divisor of f{x) and ^x) is equal to the subscript of the first of the subresultants Bq = B, B^, B^, ■■■which does not vanish. Theorem 2. If i is the degree of the greatest common divisor of two polynomials f (J) and (^{x), then this greatest common divisor may he qhtained from the ith suhresultant off and ^ by replacing the last 198 INTRODUCTION TO HIGHER ALGEBRA element in the last roxo of coefficients off by f{x), the element Just above this by xf{x), the element above this by 3?f{x), etc.; and replacing the last element in the first row of coefficients of ^ by ^(x), the element below this by xcftix), etc. If, for instance, the degrees of / and ^ are 5 and 3 respectively, and i = 1, the greatest common divisor is K »2 h h J, a^ h h h 5, h xf{x) <ji{x) x<f)(x) x^<^{x^ a:?<p{x) Consider the 70. Common Roots of Equations. Elimination. equations f(^x) = a^x" + a^x"-'^ + . . . + a„ = <f>{x) = baX^ + bj^x^'--^+-- + b^=0 J(,#0, whose roots are (Xj, a^,---a„ and /Sj, ^^,---^^, respectively; and suppose /(a;) and <l>(x) resolved into their linear factors: f{x) = «(, (a; - aj^x -a^).--{x- a„), <f>{x) = b,{x - 0^){x -^^)-{x- ^J. Since, by Theorem 1, § 65, these sets of factors are unique, it is evident that the equations /( 2;)= and 4'{x)= will have a common root when, and only when, f{x) and (p{x) have a common factor, that is, when, and only when, the resultant i2 of/ and (j> is zero. To eliminate x between two equations f{x) = and (/>(a;) = 0, is often taken in elementary algebra to mean : to find a relation between the coefficients of/ and (f> which must hold if the two equations are both satisfied; that is, to find a necessary condition for the two equations to have a common root. For most purposes, however, when we eliminate we want a relation between the coefficients which not only holds when the two equations have a common root, but such that, conversely, when it holds the equations will have a common root. From this broader point of view, to eliminate x between two equations f{x) = and <f)(x) — means simply to find a necessary and sufficient condition that these equations have a common root. Hence the result of this elimination is J? = 0. Let us, however, look at this question from a little different point of view. FACTORS OF POLYNOMIALS IN ONE VARIABLE 199 In the equations (1) af,3? + aja^ + a^+as3^+a^x+a^ = aQ=^0, (2) b^x^ + b^z-' +b^x+b^ = 5„ ^ 0, let us consider the different powers of x as so many distinct un- knowns. We have, then, two non-homogeneous, linear equations in the five unknowns x, x\ a?, a;*, x^. ■ Multiplying (1) through by x and then by a^, and (2) by x, x^, a?, a;*, in turn, we have a^'' + a-^x^ + a^ + a^x^ + a^ -f aga;^ = 0, a^ -t- a-^o^ -f a^y^ -f- a^^ -f a^ ■\- a^x = 0, a^afi + flja;*- + a^x^ + a^x^ -\- a^x + a^ =0, b^a^+b^x^+b^x+bg =0, b^ + Sj^s + b^x^ + b^x = 0, h^ H- b-^x^ -f b^ofi -t- 632^ = 0, l^x^ -f h^T> + b^xf^ 4- Jga^ = 0, h^ 4- Jja^ -F ^22^ + Sga;* = 0, a system of eight non- homogeneous, linear equations in seven unknowns. If a value of x satisfies both (1) and (2), it will evidently satisfy all the above equations. These equations are therefore consistent, so that by Theorem 1, § 16, we have. h h h h H «5 H «4 H H «3 «4 a, h h h b h h h b. h h ' = 0. Hence the vanishing of this determinant is a necessary condition for (1) and (2) to have a common root. This device is known as Sylvester's Dialytio Method of Elimina- tion.* * For the sake of simplicity we have taken the special case where ra = 5 and m = 3. The method, however, is perfectly general. 200 INTRODUCTION TO HIGHER ALGEBRA The above determinant is seen to be exactly the same as the resultant R of (1) and (2), so that Sylvester's method leads to the same condition for two epilations to have a common root as that found above, namely ^ = 0. It does not prove, however, that this condition is sufficient, but merely that it is necessary. Thus Syl- vester's method, while brief, is very imperfect. The number of roots common to two equations, fix) = and ^(x)= 0, and also an equation for computing the common roots, may be obtained at once from § 69. 71. The Cases Oq = and b^ = 0- It is important for us to note that according to the definitions we have given, the determinant 22 ( 0' "" " ) will be the resultant of the two polynomials \0n, ■ • • 0™/ fix) = aax" + a^Tf-^ +•••+«„, ^x) = \x^->r \x^-^ + ■ • • + 5^, only when / and ^ are precisely of degrees n and m respectively, that is, only when a^4=^, i^^O- Thus, for instance, the resultant of the polynomials j.^^^ ^ ^^^n-i + «^^.-2 + . . . + «^, (fix) = Jo^;™ + 5ia:™-i +--- + K, is not the (m + w)-rowed determinant i2('*i' "■'"'•) but, if a^=^0, Bq #= 0, the {m + n — l)-rowed determinant Mf i' ' " ' ^" J or, if a^ or }» is zero, a determinant of still lower order.* V c '■" m' Let us indicate by R the (m + M)-rowed determinant -K ( "' and by r the resultant of / and ^, and consider the case a^ = 0, fflj =5*= 0, b^^ 0. Since every element of the first column of R except the last is zero, we may write In a similar way we see that if the degree of / is w — i, and 6^ gfc 0, we may write ^^^jf^^ and if the degree of is m — i, and a^^O, we have R = air. Accordingly, except when a^ = J^ = 0, R differs from r only by a non-vanishing factor. * As an illustration take the two polynomials f(x) = {a. + p')x'^ + x — p and (pQi:) =ax + l. If a 4-/3^0 and a^O, the resultant here is (a'^— l)/3. But if B = — ^ :^ 0, the resultant is 1 — a^. FACTORS OF POLYNOMIALS IN ONE VARIABLE 201 Theorem. Although B T^o' " • • <\ is the resultant off and <f> onl^ when Uq^O and b^^O (or when m = or w=0), nevertheless its vanishing still forms a necessary and sufficient condition that f and (j) have a common factor even when aQ = Oorb^= 0, provided merely that both ftg and b^ are not zero. That this last restriction can not be removed is at once evident; for, if ao=6o=0, every element in the first column of the determinant is zero, and hence the determinant vanishes irrespective of whether /and have a common factor or not.* All that we can say, if we do not wish to make this exception, is, therefore, that in all cases the vanishing of R forms a necessary condition that / and ^ have a common factor. 72. The Resultant of Two Binary Forms. Let us now consider the binary forms /(a^i, x^ = a^xl + a-^xl-\ + ■■■ + a„xl (w^l) ^(^1' x^)=b,x^ + b^x^-^x^+ ... +b^x^ (jn^l). By the side of these forms we write the polynomials in one variable j<(^^-^ = a,x- + a^x-^ +•••+«„, The determinant ^f^o' •" ««] will be the resultant of F and ^ only when neither a^ nor h^ is zero. We will, however, call it the resultant of the binary forms / and <^ in all cases. * By looking at the question from the side of the theory of common roots of two equations (cf. § 70), and by introducing the conception of infinite roots, we may avoid even this last exception. An equation Ci^" + aioff^^ + — |-a„=0 has n roots, distinct or coincident, provided ao gfc 0. If we allow Oq to approach the value zero, one or more of these roots becomes in absolute value larger and larger, as is seen by the transformation x' = 1/x. Hence it is natural to say that if a(, = the equation has an infinite root. If then we consider two equations each of which has an infinite root as having a common root, we may say : A necessary and sufficient condition that the equations OoK" + aiK'-i H 1-0!„=0 M>0, 6(ia;'" + 6iX'»-i + "-+6m = m>0, 202 INTRODUCTION TO HIGHER ALGEBRA Theorem. A necessary and sufficient condition for two binary forms to have a common factor other than a constant is that their resultant be zero. If fflj and Jj are both different from zero, the non-homogeneous polynomials F and $ correspond to the forms / and ^ according ta the definition of § 62. Accordingly, by Theorem 2 of that section- a necessary and sufficient condition that / and ^ have a, common factor other than a constant is, in this case, the vanishing of their resultant. On the other hand, if a^ = J^ = 0, / and ^ have the common factor aTj, and the resultant of/ and (/> obviously vanishes. A similar renjark applies to the case in which all the a's or all the Vs. are zero. There remain then only the following two cases to be considered, (1) «„=,!= ; 5(,= Ji= ... =6^=0, 64+1 9fcO (Jc<m), (2) ifl^^; aQ = a.^= ■■• = a^, = 0, a^+i ^t {k<n). In Case (1), F corresponds to /, and, if we write ^ corresponds to <f)y Now we know in this case (cf. § 71) that M^O is a necessary and sufficient condition that F and ^ be rela- tively prime. Accordingly, by Theorem 2, § 62, it is also a necessary and sufficient condition that /and <p^ be relatively prime. But since X2 is not a factor of/, the two forms /and <f) will be relatively prime when and only when/ and (^^ are relatively prime. Thus our theo- rem is proved in this case. The proof in Case (2) is precisely similar to that just given. CHAPTER XVI FACTORS OF POLYNOMIALS IN TWO OR MORE VARIABLES 73. Factors Involving only One Variable of Polynomials in Two Variables. We have seen in the last chapter that polynomials in one variable are always reducible when they are of degree higher than the first. Polynomials in two, or more, variables are, in gen- eral, not reducible, as we have already noticed in the special case of quadratic forms. Let f{x, y) be any polynomial in two variables, and suppose it arranged according to powers of x, /(a;,y)=ao(«/>"-l-ai(t/)a:»-i+ - +a„_i(y)2;-f a„(y), the a's being polynomials in y. Theorem 1. A necessary and sufficient condition that a poly- nomial in y alone, -^y), he a factor of f{x, y) is that it be a factor of all the d's. The condition is clearly sufficient. To prove that it is necessary, let us suppose that ■^y) is a factor of /(a;, y). Then a) a^{y)x-+- + «„(2/) = ^(2/)[io(y)*"+ - +5«(2/)]' where the J's are polynomials in y. For any particular value of y we deduce from (1), which is then an identity in x, the following equations: , / ni / n ■a'n{y) = f{y)K{y)- Since these equations hold for every value of y, they are identities, and -^(y) is a factor of all the a's. 203 204 . INTRODUCTION TO HIGHER ALGEBRA Theorem 2. If f{x,y) and 4>{x, y) are any two polynbrniaU in x and y, and yjr{y) is an irreducible polynomial in y alone * which is a factor of the product fj), then ■\}r is a? factor off or of (j). Let f{x, y) = a^{y)x^+a^{y)7f--^+ - + a^{y), and <i>{x, y)^h^{y)x^ + ii{y)x"'-^+ •■• +b^{yy, then f{x, y)<i>(x, y) = a.h^x-*'- + (a,b, + a,b,)x-+^^ In order to prove that i|r is a factor either of / or of (j> we must prove that it is either a factor of all the a's or of all the b's. If this were not the case, we could find a first a in the sequence a^, a^, ■■■ a„ of which yjr is not a factor. Call this function a^. There would also be a first b in the sequence of b^, Sj, • ■ • b^ which is not divisible by tfr. Call this function bj. Our theorem will be proved if we can show that this assumption, that a^ and Jy are not divisible by yjr while all the functions a,), • ■ • a^ _ j, b^, ■■■ bj_ j, are divisible by yfr, leads to a contradiction. For this purpose let us consider in the product f(f> the coefficient of a;('»-«)+('»-J'), which may be written provided we agree that the a's and 6's with subscripts greater than n and m respectively shall be identically zero. Since f(f> is by hypothesis divisible by yjr, it follows from Theorem 1 that the last written expression must be divisible by ylr. This being obviously the case for all the terms which preceed and for all which succeed the term a^bJ, it follows that this term must also be divisible by yfr, so that among the linear factors of the function a^bj must be found yjr. But by Theorem 1, § 65, the function afij can be resolved into its linear factors in essentially only one way, and one way of so re- solving it is to resolve «< and bj into their linear factors. Since yjr is not one of these factors, we are led to a contradiction, and our theorem is proved. An important corollary of our theorem is : Corollary. Letf{x, y) and (j) [x, y) be polynomials in {x,y), and let yjr(y) be a polynomial in y alone. If yjr is a factor of the product of f^ but is relatively prime to ^, then yjr is a factor off. * That is, a linear polynoiiial- FACTORS OF POLYNOMIALS IN TWO VARIABLES 205 If yjr is irreducible, this corollary is identical with the theorem. Let us suppose yfr resolved into its irreducible factors none of which are constants, that is, into its factors of the first degree : Now consider the identity which expresses the fact that -^ is a factor (2) f{x, y)j>{x, y) = ir^{y)ir^{y) ■ ■ ■ f,{y)a{x, y). This shows that ■^i{y) is a factor of f<f) and hence, by Theorem 2, it is a factor either of / or of (f). Since yjr and ^ are relatively prime, yjr^ cannot be a factor of (j). It must then be a factor of / : f{x, y) = f^(y)fj{x, y). Substituting this in (2), and cancelling out i/r^, as we have a right to do since it is not identically zero, we get (8) f^{x, y) (f>{x, y) = y^j^y) ■ ■ ■ fly) a{x, y). From this we infer that yjr^, being a factor oi fi<f>, must be a factor ° ^' fii^^ y)=-^-iy)f-lp^ y)- We substitute this in (3) and cancel out i/rg. Proceeding in this way ^^ ^^* f{x, y) = iriiy)f2(y) ■■■ t*(y)/*(^> y)^^{y)M^, y), an identity which proves our corollary. EXERCISE If /(t, y) and tf}(x, y) are polynomials, then any two sets of polynomials Pi(.y), Qi(^. y), -Ri(^. 2'). -PjCy). Q^C^' y)' R2(^' y)' win be proportional to each other provided, (a) they satisfy the identities P^{y)f(?<:, y)^Qi(^' y)<t>(^' y) +^i(^. y)? PMf(^> y) = ^2(^' J')<^(^' y^ + ^^^''^ y^ ' (6) there is no factor other than a constant common to Pi, Qi, and also no factor other than a constant common to P^, Q^ ; (c) iJj and R^ are both of lower degree in x than <}>. (Cf. Theorem 2, § 63.) ' 206 INTRODUCTION TO HIGHER ALGEBRA 74. The Algorithm of the Greatest Common Divisor for Polyno- mials in Two Variables. We will consider the two polynomials in X and y, ^^^^ ^^ ^ ^^^^^^„ ^ ^^^^^^„_i ^ _^ ^j^^^^ <\>{x, y) =6o(2/K + *i(yK~' + - + Uy> and assume a^ ^ 0, 6q ^ 0, w ^ m >0. Theorem 1 of the last section in combination with the results of § 67 enables us to get all the common factors of / and <^ which in- volve y only; for such factors must be common factors of all the as and all the 6's. It remains, then, merely to devise a method of obtaining the com mon factors of/ and ^ which do not themselves contain factors in y alone. We v,ill show how this can be done by means of the algo- rithm of the greatest common divisor. Dividing / by (^ (cf . § 63, Theorem 2), we get the identity -Po(s')/(^' y)= ^o(^' ^^)^(^' y) + ^i(^' y)^ when iZj is either identically zero, or is of lower degree in x than ^. If R^ ^ 0, divide ^ by R^, getting the identity Pi{y)'l>{^^ y) = ^i(«' y)^i(.^^ y) + ^i{^^ «/> where R^ is either identically zero, or is of lower degree in x than i2j. If R^ ^0, divide R^ by R^ Proceeding in this way, we get the following system of identities in which the degrees in x of R-^, R^, ■■■ continually decrease, so that after a certain number of steps we reach an R, say -Bp+j, which is independent of x : -Po(2')/(^' y) = ^o(^' y)<t>{^^ y) + -^i(^' y)^ Pi(.y)4'(^^ y) = Qiip^ y)^i(*' y) + ^2(^^ y)^ A(2/)-^i(^' y) = Qii^^ y)R-i{?>-> y) + -^3(2;, y). (1) P9-i{yW,--i(p^ y) = Qp-ii^^ y)Rp-i(^^ y)+ ^-l^^ y> P,{y)R,-i{^^ y) ^ Q,{^-^ ^)^p(^, y) + RUy)- Theoeem 1. A necessary and sufficient condition thatf and <}> have a common factor which involves x is FACTORS OF POLYNOMIALS IN TWO VARIABLES 201 In order to prove this theorem we first note that, by the first of the identities (1), any common factor of/ and </> is a factor of R^, hence, by the second of the identities, it is a factor of R^, etc. Finally we see that every common factor off and <j) is a factor of all the Ji's. But Rp+i does not contain x. Hence if /and <f> have a common factor which contains x, Rp+i{y) = 0. Now suppose conversely that ^p+j(«/)=0, and let (2) B,{x,y)^S{y)a(x,y), where Gr has no factor in y alone.* The last identity (1) then tells us that -Pp(«/) is a factor of 'Qlx, y)S{y)a(x, y), and since by hypothesis Gr has no factor in y alone, PJi^y) must, by the Corollary of Theorem 2, § 73, be a factor of Q^S, that is (3) Q,{x, y)S(y) = P,{y)Eix, y). Substituting first (2) and then (3) in the last identity (1), and cancel- ling out the factor Pp(y) from the resulting identity, as we have a right to do since Pp{y) ^ 0, we get the result ^P-ii^^ I/)^S(x, y)G-{x, y). That is, (r is a factor not only of R,, but also of R;,-^. Accordingly we may write the next to the last identity (1) in the form ^p-i(«/)-^^-2(^' y')=J''K^- y)^{^-. y)- By the corollary of Theorem 2, § 73, we' see that Pp-i{y) is a factor of .T", so that Pp-i(y) can be cancelled out of this last written identity, and we see that 6r is a factor of -Bp.g. Proceeding in this way, we see that Q^ is a factor of all the J2's, and therefore, finally, of / and <f>. Moreover, we see from (2) that (7 is of at least the first degree in x, as otherwise Rp would not con- tain X, while Rp+i was assumed to be the first of the ^'s which did not involve x. Thus our theorem is proved. Since, as we saw above, every common factor of / and 4> is also a factor of all the R's, it follows from (2) that, if i/r is a common factor of/ and <^, (3L(^^ y^s{y) = f{x, y)K{x, y). * If Bp has no factor in y alone, S reduces to a constant. 208 INTRODUCTION TO HIGHER ALGEBRA If then yjr contains no factor in 1/ alone, S must, by the Corollary of Theorem 2, § 73, be a factor of K. Consequently by cancelling out S from the last written identity, we see that -^ is a factor of Gr. That is, Theorem 2. If in Euclid's algorithm -Bp+i = 0, the greatest cem- mon divisor of f and <f> which contains no factor in y alone is the poly- nomial Gr{x,y) obtained by striking from Rf,{x,y) all factors in y alone. We note that if R^+i is a constant different from zero, / and ^ are relatively prime ; but that the converse of this is not true as the simple example f = 2x^ + 2.y\ if> = x shows. Going back to the identities (1), we get from the first of these identities, by mere transposition, the value of R^ in terms of / and ^ (and Pq, §q). Substituting this value in the second identity, we get a value for R^ in terms of /, (^, and certain P's and Q's. Pro- ceeding in this way, we finally get the formula (4) ^P+i(«/) = Fix, y)f(x, y) + ^{x, y)^{x, y) where I' and ^ are polynomials in (a;, y). 75. Factors of Polynomials in Two Variables. Theorem 1. If fix, y) and ^(a;, y) are any two polynomials in x and y, and yjrix, y) is an irreducible polynomial which is a factor of the product f(f), then yjr is a factor of f or of (f). If i/r does not contain both x and y, this theorem reduces to Theorem 2, § 73. It remains, then, only to consider the case that yjr involves both variables. In this case, at least one of the polynomials /, <f> must be of at least the first degree in x. Without loss of gener- ality we may assume this to be/. If i|r is a factor of/, our theorem is true. Suppose i/^ is not a factor of/; then, since i|r is irreducible, / and i/r are relatively prime, and if we apply the algorithm of the greatest common divisor to /and ifr (as we did in the last section to / and 0) the first remainder R^^^ (y) which does not involve x is not identically zero- The identity (4) of the last section now becomes (1) i2.+i(y) = Fix, y)fix, y) + ^ix, y)ylr{x, y). FACTORS OF POLYNOMIALS IN TWO VARIABLES 209 If we multiply this by <J3{x, y), the second member becomes a poly- nomial which has i/r as a factor, since, by hypothesis, /^ has •^ as a factor. We may therefore write C^) ^p+i {y)K^', y) = -^{x, y)x{x, y). Now no factor other than a constant of i2p+j can be a factor of i/r, since -^ is irreducible. Consequently, by the Corollary of Theorem 2, § 73, ^p+j is a factor of ■)^x, y). Cancelling out B^^^ from (2), as we have a right to do since it does not vanish identically, we get an identity of the form <l){x, y) = '>lr{x, y)xi{x, y); that is, i/r is a factor of (^, and our theorem is proved. By applying this theorem a number of times, we get the CoKOLLAKY. If the product of any numher of polynomials in two variables/ A{x, y)f,(x, y) ■ ■ ■ f^x, y), is divisible by an irreducible polynomial 1/^(2;, y), then yfr is a factor of at least one ofthefs. We come now to the fundamental theorem of the whole subject of divisibility of polynomials in two variables. Theorem 2. A polynomial in two variables which is not identically zero can be resolved into the product of irreducible factors no one of which is a constant in one, and essentially in only one, way. That a polynomial f{x, y) can be resolved into the product of irreducible factors no one of which is a constant in at least one way may be seen as follows. If /is irreducible, no factoring is possible or necessary. If/ is reducible, we have f{x,y)=flx,y)flx,y), where neither /j nor/g is a constant. If /^ and f^ are both irredu- cible, we have a resolution of/ of the form demanded. If not, resolve such of these polynomials /j and/g as are reducible into the product of two factors neither of which is a constant. We thus get / ex- pressed as the product of three or four factors. This is the resolu- tion of/ demanded if all the factors are irreducible. If not, resolve such as are reducible into the product of two factors, etc. This pro- cess must stop after a finite number of steps, for each time we factor 210 INTRODUCTION TO HIGHER ALGEBRA a polynomial into two factors, the degrees of the factors are lowei than the degree of the original polynomial. We shall thus ulti- mately resolve / by this process into the product of irreducible factors, no one of which is a constant. Suppose now that / can be resolved in two ways into the product of irreducible factors none of which are constants, f{x, y)=A{x, y)f^{x, y) ■■■fk{x, y) = ^^{x,y)<p^{x,y)--<^i{x,y). Since c^j is a factor of /, it must, by the Corollary of Theorem 1, be a factor of one of the polynomials /ji/gi "' fk- Suppose the/'s so arranged that it is a factor oify Then, since /j is irreducible, /j and ^j can differ only by a constant factor, and since (p-^^O, we may cancel it from the identity above, getting In the same way we see from this identity that/2 and one of the <t's, say (^2' differ only by a constant factor. Cancelling ip^, we get ''I'^z/s ■•■fk = 4>3 •■■ ^i- Proceeding in this way, we should use up the <p's before the /'s if Kk, the/'s before the <f>'s ii I >k. Neither of these cases is possible, for we should then have ultimately a constant on one side of the identity, and a polynomial different from a constant on the other. Thus we must have k = I. Moreover we see that the /'s can be arranged in such an order that each / is proportional to the corre- sponding c6, and this is what we mean (cf. Definition 7, § 60) by saying that the two methods of factoring are not essentially different. Thus our theorem is proved. Theorem 3. If two polynomials f and <f> in {x, y') are relatively prime, there are only a finite number of pairs of values of (x, y) for which f and (j> both vanish.* For if/ and (f) both vanished at the points (3) (^v 2/1)' (^2' 2/2)' ••• ' and if these points were infinite in number, there would be among them either an infinite number of distinct x's or an infinite number of * stated geometrically, this theorem tells us that two algebraic plane curves f(z, y) = 0, <j>(x, j/) = Ocan intersect in an infinite number of points only when thej have an entire algebraic curve in common. FACTOKS OF POLYNOMIALS IN TWO VARIABLES 211 distinct «/'s. By a suitable choice of notation we may suppose that there are an infinite number of distinct y's. Then it is clear that ;f and ^ must be of at least the first degree in x, since a polynomial in y alone which does not vanish identically cannot vanish for an infinite number of values of y. We may then apply to / and ^ the algorithm of the greatest common divisor as in § 74, thus getting (cf. (4), § 74) an identity of the form (4) F{x, yy{x, y) + <^{x, y)4>{x, y) = R^^,(y) m 0- Since the first member of (4) vanishes at all the points (3), R^+^y) would vanish for an infinite number of distinct values of y, and this is impossible. An important corollary of the theorem just proved is that if / and ^ are two irreducible polynomials in {x, «/), and if the equations /"= and ^=0 have the same locue, then/ and <^ differ merely by a constant factor. This would, however, no longer be necessarily true if/ and were not irreducible, as the example, /=2;«/2, <l> = x^y, shows; for the two curves /=0 and <^ = are here identical, since the curve in each case consists of the two coordinate axes, and yet/ and ^ are not proportional. By means of the following convention, however, the statement made above becomes true in all cases : Let/ be resolved into its irreducible factors, where /i, ■■■f^ are irreducible polynomials in {x, y), no two of which are proportional to each other. The curve /= then consists of the A pieces, ^^^ 0,/2 = 0, ••■/4 = 0. To each of these pieces we attach the corresponding positive integer «^ which we call the multiplicity of this piece ; and we then regard two curves given by algebraic equations as identical only when they consist of the same irreducible pieces, and each of these pieces has the same multiplicity in both cases. With this convention we may say : Corollary. Iff and ^ are polynomials in (x, y) neither of which is identically zero, a necessary and sufficient condition that the two curves /= 0, ^ = 5e identical is that the polynomials f and 4> differ only by «. constant fact^" 212 INTRODUCTION TO HIGHER ALGEBRA EXERCISES 1. Let/(x), <^(a;), i/'(a;) be polynomials in i whose coefficients lie in a certain domain of rationality. Then if i^ is irreducible in this domain and is a factor of the product /<^, prove that kj/is a, factor of /or of <!>. 2. Let/(a;) be a polynomial in x, which is not identically zero, and whose coefficients lie in a certain domain of rationality. Prove that /can be resolved into a product of polynomials whose coefficients lie in this domain, which are ■^ irreducible in this domain, and no one of which is a constant, in one and essen- tially in only one way. 3. Extend the results of this section to polynomials in two variables whose coefficients lie in a certain domain of rationality. 76. Factors of Polynomials in Three or More Variables. The re- sults so far obtained in this chapter may be extended to polynomials in three variables without, in -the main, essentially modifying the methods already used. We proceed therefore to state the theorems in the order in which they should be proved, leaving the proofs of most of them to the reader. The extension to the case of n variables then presents no difficulty, and is left entirely to the reader (cf. Exercise 1). Let f{x, y, z) be any polynomial in three variables, and suppose it arranged according to powers of x, f(x, y, z) = a^^y, s)a;" + a^{y, 2)a:»-i + - + a„(y, z), the a's being polynomials in (y, z). Corresponding to Theorems 1, 2 of § 73 we have Theorem 1. A necessary and sufficient condition that a polynomial in (y, z) he a factor off is that it be a factor of all the a's. Theorem 2. If f{x, y, z) and <fi{x, y, z) are any two polyno- mials in {x, y, z) and ■>p-{y, z) is an irreducible polynomial in {y, z) only which is a factor of the product f^, then ■^ is a factor offor of (j). COEOLLAEY. Let f(x, y, z) and (j>(x, y, z) be polynomials in {x, y, z), and let ■\)f{y, z) be a polynomial in {y, z) alone. If i^ is a factor of the product off(j),but is relatively prime to (f), then it is a fac- tor off. To find the greatest common divisor of two polynomials in three variables we proceed ''xactly as in the case of two variables, getting FACTORS OF POLYNOMIALS IN TWO VARIABLES 213 a set of identities similar to (1), § 74, the P's and -Bp+j being now functions of {y, z), while the other ^'s and the ^'s are functions of {x, y, z). Corresponding to Theorems 1, 2 of § 74 we now have Theokem 3. A necessary and sufficient condition thatf{x, y, z), and (f>{x,y, z) have a common factor which involves x is that Ili,_^-j^{y, z)s 0. Theorem 4. If Hp+^^y, z)=0, the greatest common divisor of f{x, y, z) and <f>(^x, y, z) which contains no factor in (_y, z) alone is the polynomial G{x, y, z) obtained by striking out from R^ix, y, z) all factors in (y, z) alone. From the algorithm of the greatest common divisor for the two polynomials /(a;, y, z), tj){x,y,z) we also deduce the identity (1) -Bp+i(2/, 2) s Fix, y, z)f{x, y,z) + ^ {x, y, z)<j)(x, y, z), similar to (4), § 74. Corresponding to Theorems 1, 2 of § 75 we have Theorem 5. If f{x, y, z) and <f>{x, y, z) are any two polynomials and'\lr{x, y, z)is an irreducible polynomial which is a factor of the prod- uct f^, then -^ is a factor of f or of <^. Corollary. If the product of any number of polynomials /i(a;. y^ 2)/2(«. y^ 2) --fki^^y^^)^ is divisible by an irreducible polynomial -\lr{x, y, z), then i/r is a factor of at least one of the f^s. Theorem 6. A polynomial in three variables which is not identi- cally zero can he resolved into the product of irreducible factors no one of which is a constant in one, and essentially in only one, way. When we come to Theorem 3, § 75, however, we find that it does not admit of immediate extension to the case of three vari- ables ; for Bp+j{y), which came into the proof of that theorem, becomes now Ii^+^{y, z), and we can no longer say that this does not vanish at an infinite number of points {y, z). Not only is the proof thus seen to fail, but the obvious extension of the theorem itself is seen to be false when we recall that two surfaces intersect, in gen- eral, in a curve. This theorem may, however, be replaced by the following one : Theorem 7. Iff{x, y, z) and 4>(x, y, z) are any two polynomials in three variables of which is irreducible, and if f vanishes at all points {x, y, z) at which <f> vanishes, then is a factor off. 214 INTRODUCTION TO HIGHER ALGEBRA In proving this theorem we may, without loss of generality, assume that <f) actually contains one of the variables, say x ; for if (^ contains none of the variables x, y, a, the theorem is trivial and obviously true. Suppose (}> were not a factor of /. Then, since cf> is irreducible, / and <^ are relatively prime. Hence, in the identity (1) above, Rp^i{y, 2) ^ 0. Let us write (2) <l>{x, y, z) = h,{y, z)x'- + b^(y, z)x^-^ +■■■ + b,,(y, z) {m> 1), where, without loss of generality, we may assume hQ^y, z) ^ 0. Then (3) B,^^{y,z)h^{y,z)^0. Accordingly we can find a point (y^, 2j) such that (4) ■«P+i(yr^i)^0, h,{y„z{)^0. Consequently <^(a;, y^, a^) is a polynomial in x alone which is of at least the first degree, and which therefore (Theorem 1, § 6) vanishes for some value x^ of x. That is <f>(x^, yy 2i) = 0. Accordingly, by hypothesis. Referring now to the identity (1), we see that -Kp+i(2/i'«i) = 0. This, however, is in contradiction with (4). Thus our theorem is proved. If to each part of a reducible algebraic surface we attach a multi- plicity in precisely the same way as was explained in the last section for plane curves, we infer at once the Corollary. Iff and cj) are polynomials in (x, y, z) neither of which is identically zero, a necessary and sufficient condition that the two sur- faces y= 0, <^ = be identical is that the polynomials f and (f> differ only by a constant factor. Theorem 7 admits also the following generalization : Theorem 8. Iff{x, y, z) and 4>{x, y, z) are any two polynomials in three variables which both vanish at the point (2;^, y^, z^) and of which <j) is irreducible, and if in the neighborhood iV of {xq, y^, z^) f vanishes at ell points at which 4> vanishes, then (p is a factor off. FACTORS OP P0LYN0MIA1.S IN TWO -VARIABLES 215 We assume, as before, that <f> contains x and can therefore be ivritten in the form (2). Let us first consider the case in which ^oC^O' ^o) ^ ^- Here the proof is very similar to the proof of Theorem 7. We obtain relation (3) precisely as above, and from it we infer that a point (i/j, z^) in as small a neighborhood M of (y^, Zg) as we please can be found at which the relations (4) are true. Now consider the equation (5) 0(a;, y^, 3^)=0. By writing ^ in the form (2). we see that by taking the neighbor- hood M oi (y^, ^o) sufficiently small, we can make the coefficients of (5) differ from the coefficients of (6) <iix, ^0, Zq)= by as little as we please (cf. Theorem 3, § 5). Now a;,, is by hy- pothesis a root of equation (6). Consequently by taking M suffi- ciently small, we can cause (5) to have at least one root x^ which differs from x^ by as little as we please (cf. Theorem 4, § 6). Thus we see that a point {x-^, y^, z^ in the given neighborhood N of {xq, y^, 2q) can be found at which ^{xy, yj, 2i)= 0. Accordingly, by hypothesis, From the identity (1) we 1-ave then ^P+i(^i'2i)=0» which is in contradiction with (4). Thus our theorem is proved on the supposition that SoC^o' ^o)''^*^-* * The proof just given vdll, in fact, apply to the case in which not all the &'s in (2) vanish at the point (j/o, Zo\ « we use the extension of Theorem 4, § 6, which is there mentioned in a footnote. It is only the extreme case in which all the 6's vanish at this point which requires the special treatment which we now proceed to give. The reader is advised to consider the geometrical meaning of this extreme case. 216 INTRODUCTION TO HIGHER ALGEBRA In order to treat the case in which Jolyo'^^o)" *'' ^®*' '^^ denote by k the degree of </)(a5, y, z), and let us subject this polynomial to a non-singular linear transformation (7) • y^a^x' + ^^y' + ^2^ 2 = a^x' + ^3^' + 78S' which makes the degree of <f) inx equal to the total degree k oi <f> (cf. Theorem 2, §64). Suppose that this transformation carries over the point (3:^,1/^,20) into the point (x^, y'^, Zq). Then it is possible, since (7) is non- singular, to take such a small neighborhood N' of (a;^, y^, 2q) that all points in this neighborhood correspond to points in the given neighborhood Ifoi {xq, y^, Zg). Moreover, by means of (7), ^ has gone over into (8) ^\x\ y', z') ^ b'ox'' + biiy', z')x''"' + ■■■ + H{y', z'), where SJ is a constant different from zero. Let us denote by f'{x', y', z') the polynomial into which / is transformed. Then it is clear that, since, in the neighborhood N,/ vanishes whenever <^ does, in the neighborhood If' (which corresponds to a part of iV"),/' vanishes whenever (j)' does. Accordingly we can apply the part of the theorem already proved to the two polynomials /' and (^', since the first coefficient of ^' in the form (8), being a constant different from zero, does not vanish at {yl), z'q). We infer that <}>' is a factor °^-^'' nx', y', z>) = <f>'(:x', y', z')ylr'(x', y>, z'). If here we replace x', y', z' by their values in terms of x, y, z from (7), we see that <j> is a, factor of /; and our theorem is proved. EXERCISES 1. State and prove the eight theorems of this section for the case of polyno- mials in n variables. 2. Extend the result of the exercise at the the end of §73 to the case of polynomials in n variables. 3. Extend the results of the two preceding exercises to the case in which we consider only polynomials whose coefficients lie in a certain domain of ration' ality. FACTORS OP POLYNOMIALS IN TWO VARIABLES 217 4. The resultant of two polynomials in one variable /(a:)=ao«» + aia;»-'+ ... + o„, ^ (x)=6„x"'+ 6ix'»-i+ - + bm, is sometimes defined as the polynomial R in the a'a and 6's of lowest degree which satisfies an identity of the form , , _ where F and * are polynomials in (oo, ■•• a,; b^, ■■■ b„; x), and the identity is an identity in all these arguments. Prove that the resultant as thus defined differs only by a constant factor different from zero fron^the resultant as we defined it in § 68. CHAPTER XVII GENERAL THEOREMS ON INTEGRAL RATIONAL INVARIANTS 77. The Invariance of the Factors of Invariants. Let as con. sider the general w-ary form of the Jcth degree which we will rep- resent by /(a;j, ••• »;„ ; aj, a^, •••), the x's being the variables and the a's the coefficients. By suitably changing the a's, this symbol may be used to .represent any such form. Hence, if we subject such a form to a linear transformation, the new form, being M-ary and of the same degree as the old, may be represented by the same func- tional letter : /(^J, •■■ os'„; a[, a'^, •••). This new form will evidently be homogeneous and linear in the a's ; that is, each of the a''s is a homogeneous linear polynomial in the a's. It is also clear that each of the a''s is a homogeneous polynomial of the Jcth degree in the coefficients of the transformation. It follows from the very definition of invariants that if we have a number of integral rational relative invariants of a form or system of forms, their product will also be an integral rational relative in- variant. It is the converse of this that we wish to prove in this section. We begin by stating this converse in the simple case of a (-ingle form. Theorem 1. If I{a-^, a^, ■••) is an integral rational invariant of the n-ary form j., _ > and is reducible, then all its factors are invariants. It will evidently be sufficient to prove that the irreducible factors of I are invariants. Let f^,f^, •••'/, be the irreducible factors of /. Subjecting /to the linear transformation ■Xi = c^^x[+ ... + Oi^x'„, ^n — "nl^l + ••• + ^n!i*»' 218 , INTEGRAL RATIONAL INVARIANTS 219 whose determinant we call o, and denoting the coefficients of the transformed form by a[, a'^, ■■■ , we have (1) I{a[, 4, ■••) = c^-fC^i, «2' •••^: an identity which may also be written We have here a polynomial in the c's and a's which, on the second side of the identity, is resolved into its irreducible factors, since by Theorem 1, §61, the determinant c is irreducible. Hence each factor on the first side is equal to the product of some of the factors on the second. That is (2) fj^a[, 4, • • •) = c^'M«i, «2' • • •) (i = 1, 2, • • • I) where the <f)'s are polynomials. Now let _ _ _ _ 1 ''ll — ''22 — ■" — ^in — ' and let all the other c's be zero. Our transformation becomes the identical transformation, the determinant e = 1, and each a' is equal to the corresponding a. The identities (2) therefore reduce to • /i(«l> «2' ■ • •) = ^I'^v «2' ■ • ■) (i = 1, 2, • • • I). Substituting this value of </>,- in (2), we see that fi is an invariant, and our theorem is proved. The general theorem, now, is the following : Theorem 2. If I (aj, a^, — ;h^,h^,--- ; ■■■) is an integral rational invariant of the system of forms fli^V ■■■ ^n> ^V ^V ■■■) and is reducible, then all its factors are invariants. The proof of this theorem is practically identical with that of Theorem 1. 220 INTRODUCTION TO HIGHER ALGEBRA EXERCISE If /(qi, 02, ■••; bi,b2, ■••; ■■■; yi, ■•• l/n', ^i. ■•■^ni "•) is an integral rational co Tariant of the system of forms /i(a:i, •••x„; 01,02,—), /zC^i, ■■•a;„; fti.fe, •■•)> and the system of points (yi,--y„),(zi,- factors are oovariants (or invariants). ■z„),---, and is reducible, then all its 78. A More General Method of Approach to the Subject of Rela- tive Invariants. We have called a polynomial I in the coefficients of an w-ary form / a relative invariant of this form if it has the property of being merely multiplied by a power of the determinant of the transformation when /is subjected to a linear transformation. It is natural to inquire what class of functions Zwe should obtain if we make the less specific demand that I be multiplied by a polj'- nomial in the coefficients of the transformation. We should expect to get in this way a class of functions more general than the invari- ants we have so far considered. As a matter of fact, we get precisely the same class of functions, as is shown by the following theorem : Theorem. Let I be a polynomial not identically zero in the co- efficients (aj, a^, •••) of an n-ary form f and let {a\, a'^^ •••) he. the co- efficients of the form obtained by subjecting f to the linear transformation If - I{a[, a^, • • •) s i/r(cjj, • • • c„„) /(aj, a^, ■■■), where i^ is a polynomial in the c's, and this is an iaentity in the d's and e's, then ■\jr is a power of the determinant of the transformation. We will first show that -^^O when c^Q. If possible let dj^, ■ • • d„.^ be a particular set of values of the c^^'s such that while dr d,, <*»1 ■ ■ • <*n» ^0. INTEGRAL RATIONAL INVARIANTS Then the transformation 221 has an inverse Let us consider a special set of a's such that I{a-^, a^, ■ • •) ^ 0. Then I{a[, a^, • ■ •) = -f (<^ir • • • <^«») -^ («r «2' • • •) = 0- Now apply the inverse transformation, and we have which is contrary to our hypothesis. Having thus proved thafi/r can vanish only when c = 0, let us break up yjr into its irreducible factors, Since yjr vanishes whenever i|r^=0, i|r^ can vanish only when c=0. Hence by the theorem for n variables which corresponds to Theorem 7^ § 76, i/r,- must be a factor of c. But e is irreducible. Hence i/r^ car differ from e only by a constant factor, and we may write f=Ke\ It remains then merely to prove that the constant ^has the value 1 For this purpose consider the identity and give to the c^^'s the values which they have in the identical transformation. Then e = 1, and the a''s are equal to the correspond' ing a's. The last written identity therefore becomes /(«!, a^, ■■■)=KI{a^,a^,--); from which we infer that K=l. 232 INTRODUCTION TO HIGHER ALGEBRA EXERCISSS 1. Prove that if a polynomial / in the coefficients ai, a^, •■• of an ra-ary form and the coordinates (j/i, ■■■yn) of a point has the property of being merely multiplied by a certain rational function \p of the coefficients of the transformation when the form and the point are subjected to a linear transformation, then i// is a positive or negative power of the determinant of the transformation, and / is a oovariant. 2. Generalize the theorem of this section to the case of invariants of a system of forms. 3. Generalize the theorem of Exercise 1 to the case of a system of forms and a system of points. 4. Prove that every rational invariant of a form or system of forms is tJie ratio of two integral rational invariants. 5. Generalize the theorem of Exercise 4 to the case of covariants. 79. The Isobaric Character of Invariants and Covariants. In many investigations, and in particular in the stud}' of invariants and covariants, it is desirable to attach a definite weight to each of the variables with which we have to deal. To a product of two or more such variables we then attach a weight equal to the sum of the weights of the factors, and this weight is supposed to remain unchanged if the product is multiplied by a constant coefficient. Thus if gj, gg, Z3 are regarded as having weights Wy, w^^ Wg respectively, the term 5 2j Z-t 2(5 Z' 3 would have the weight w-^ + w^ + 2w^. If, then, having thus attached a definite weight to each of the vari- ables, we consider a polynomial, each term of this polynomial will be of a- definite weight, and hy the weight of a polynomial we understand the greatest weight of any of its terms whose coefficient is not zero. If moreover all the terms of a polynomial are of the same weight, ths polynomial is said to be isobaric. It may be noticed that, according to this definition, a polynomial which vanishes identically is the only one which has no weight, while a polynomial which reduces to a constant different from zero is of weight zero. Moreover if two polynomials are of weights Wj and w^, their product is of weight Wj + w^.* * The conception of degree of a polynomial is merely the special case of the con- ception of weight in which all the variables are supposed to have weight 1. The con. ception of being isobaric then reduces to the conception of homogeneity. INTEGRAL RATIONAL INVARIANTS 223 We will apply this conception of weight first to the case in which the variables of which we have been speaking are the coefficients a-^, a^, ••■ of the n-avj form j\^ii ■■■*»; (^\i *2' ■■■)• We shall find it desirable to admit n different determinations of the weights of these a's ; one determination corresponding to each of the variables a;^, • • • x^. Definition 1. If cii is the coefficient of the term in an n-ary form, we assign to a^ the weights PitP^i"- Pn respectively with regard to the variables Xp x^, ■■■ x„. In the case of a binary form, ^0*^1 ' ^1*^1 '^2 t "' * ^A'^2' the subscripts of the coefficients indicate their weights with regard to a^g, while their weights with regard to x-^ are equal to the differences between the degree of the form and these subscripts. As a second example, we mention the quadratic form n ^a^jX^Xji Here the weight of any coefficient with regard to one of the vari- ables, say xp is equal to the number of times j occurs as a subscript to this coefficient.* In connection with this subject of weight, the special linear transformation , /. ... .^. {Xi = x[ {i=^j) Xj = kx'j is useful. If Ui is a coefficient which is of weight \ with regard to X; the term in which this coefficient occurs contains x), and therefore * For forms of higher degree, a similar notation for the coeflScients by means of multiple subscripts might be used. The weight of each coefficient could then be at once read ofi from the subscripts. 224 INTRODUCTION TO HIGHER ALGEBRA That is Theorem 1. The weight with regard to Xj of a coefficient of an n-ary form is the exponent of the power of k by which this coefficient is multiplied after the special transformation (1). From this it follows at once that an isobaric polynomial of weight X, with regard to Xj in the coefficients (aj, a^, ■■•) of an w-ary form is simply multiplied by k'' if the form is subjected to the linear transformation (1). Moreover, the converse of this is also true. For if aj, a'^,--- are the coefficients of the w-ary form after the transformation (1), and if ^(aj^, ag, •••)is a polynomial which has the property that this being an identity in the a's and also in k, we can infer, as fol- lows, that (^ is isobaric of weight X. Let us group the terms of ^ together according to their weights, thus writing <^ in the form ^(ar«2' •■•) = '^i(«i'«2' ■••) + ^2(«i'«2' •■■)+ ••• where </>i, c^j, • ■ ■ are isobaric of weights Xj, \^, ■■■ . We have then But on the other hand <#>(«'r«2' •■■) = ^</'(«i'«2' •••)sF(/>i(ai, flj, ■■■) + k>'cl>^(a^,a^, ...)+.... Comparing the last members of these two identities, we see that A. = A-j ^= An = ■ • • as was to be proved. We have thus established the theorem : Theorem 2. A necessary and sufficient condition that a poly- nomial (f) in the coefficients of an n-ary form be simply multiplied by P when the form is subjected to transformations of the form (1) is that 4> be isobaric of weight \ with regard to Xj. By means of this theorem we can show that the use of the word weight introduced in §31 is in accord with the definition given in the present section. For an integral rational invariant of an w-ary form which, according to the definition of §31, is of weight \ will, if IKTEGRAL RATIONAL INVARIANTS 225 the form is subjected to the transformation (1), be merely multiplied by k'' and must therefore, according to Theorem 2, be isobaric of weight \ with regard to Xj. That is: Theorem 3. Tf I is an integral rational invariant of a form f which according to the definition of § 31 is of weight X, it will also be of weight \ with regard to each of the variables Xj of f according to the definitions,of this section, and it will be isobaric with regard to each of these variables. As an illustration of this theorem we may mention the discrimi- nant a^a^ _ a2 of the binary quadratic form a^xl + 2a-^x-^X2 + a^x^ which is isobaric of weight 2 both with regard to Xi and with regard to X2. The reader should consider in the same way the discriminant of the general quadratic form. All of the considerations of the present section may be extended immediately to the case in which we have to deal, not with a single form, but with a system of forms. We state here merely the theorem which corresponds to Theorem 3. Theobbm 4. If I is an integral rational invariant of a system of forms which according to the definition of § 31 is of weight X, it will also be of weight X with regard to each of the variables Xj of the system, and it will be isobaric with regard to each of these variables. The reader may consider as an illustration of this theorem the resultant of a system of linear forms, and also the invariants obtained in Chapters XII and XIII. We saw in Theorem 5, § 31, that the weight of an integral rational invariant cannot be negative. This fact now becomes still more evident, since the weight of no coefficient is negative. Moreover, we can now add the following further fact: Theorem 5. An integral rational invariant of a form or system of forms cannot be of weight zero. For consider any term of the invariant whose coefficient is not zero. This term involves the product of a number of coefficients of the system of forms. Since none of these coefficients can be of nega- Q 226 INTRODUCTION TO HIGHER ALGEBRA tive weight, the weight of the term will be at least as great as the weight of any one of them. But any one of them is at least of weight 1 with regard to some one of the variables. Hence the in- variant is at least of weight 1 with regard to. some one of the variables, and hence with regard to any of the variables. In order, finally, to be able to extend the considerations of this section to the case of covariants, we must lay down the following additional definition: Definition 2. If the sets of variables (j/j, ••• y„), (z^, ••■ «„), ■•• are cogredient with the variables (a^j, ••• x^ of a system of n-ary forms, we will assign to yj, z^ ■ ■ ■ the weight — 1 with regard to xp to all the other y's, g's, etc. the weight 0. It will be noticed that here too, when we perform the transform- ation (1), each of the variables is multiplied by a power of k whose exponent is the weight of the variable. It is therefore easy* to extend the considerations of this section to this case, and w^e thus get the theorem : Theorem 6. If I is an integral rational covariant of a system of forms and a system of points which is of weight X according to the definition q/" § 3 1, it will also be of weight X with regard to each of the variables of the system, and it will be isobario with regard to each of these variables. As an example of this theorem we note that the polar of a binary quadratic form is isobaric of weight zero. The reader may satisfy himself that the same is true of the polar of the general quadratic form. 80. Geometric Properties and the Principle of Homogeneity It is a familiar fact that many geometric properties of plane curves or surfaces are expressed by the vanishing of an integral rational func- tion of the coefficients of their equations. Take, for instance, the surface ^1) /(a:, y, 3; «!, «2' •••) = 0, * Slight additional care must be taken here on account of the possible presence of terms of negative weight. mTEGRAL RATIONAL INVARIANTS 22'i where / is a polynomial of the ^th degree in the non-homogeneoua cooidinates a;, y, z, and asj, a^,--- are the coefficients of this polyno- mia!l ; and consider the relation 12) </,(«!, «„...) = 0, where <^ is a polynomial, which we will assume to be of at least the first degree, in the coefficients asj, a^-. ■■• . By Theorem 3, § 6, there are an infinite number of polynomials of the kih. degree in {x, «/, s) whose coefficients satisfy the relation (2) and also an infinite number whose coefficients do not satisfy this relation. In other words, all polynomials of the 4th degree in (x, y, z) may be divided into two classes, A and B, of which the first is characterized by con- dition (2) being fulfilled, while the second is characterized by this condition not being fulfilled. We may therefore say that (2) is a necessary and sufficient condition that / have a certain property, namely, the property of belonging in class A. The simplest examples, however, show that this property of / need not correspond to a geometric property of the surface (1). To illustrate this, let k=l, so that we have /= a^x + a^y + a^z + a^. And consider first the polynomial in the a's : The vanishing of <j) gives a necessary and sufficient condition that / belong to the class of homogeneous polynomials of the first degree in (x, y, z), and thus expresses a property of the polynomial. This same condition, a^ = 0, also expresses a property of the plane/= 0, namely, the property that it pass through the origin. Suppose, however, that instead of the function-^ we take the polynomial ^^ = «^_1. The vanishing of this polynomial also gives a necessary and suffi cient condition that the polynomial / have a cei-tain property, namely, that its constant term have the value 1. It does not serve to dis tinguish planes into tvro classes, since we may write the equation of any plane (except those through the origin) either with the constant term 1 or with the constant term different from 1 by merely multi plying the equation through by a constant. 228 INTRODUCTION TO HIGHER ALGEBRA From the foregoing it will be seen that saying that a surface has a certain property amounts to the same thing as saying that it belongs to a certain class of surfaces.* Theorem 1. The equation (2) expresses a necessary and sufficient condition for a geometric property of the surface (1) when, and only when, the polynomial (f) is homogeneous. For if <}) is non-homogeneous, let us wilie it in the form <^=^« + ^„-l^ H^i + </>o' where (f)„ is a homogeneous polynomial of the nth degree and each of the other 0's which is not identically zero is a homogeneous poly- nomial of the degree indicated by its subscript. Let a[, a'^, ■ ■ ■ he ?^ set of values of the a's for which (}>„ and at least one of the other <^/s is not zero, and consider the surface (3) f(x, y, s; cd^, ca\, ••■) = 0. The condition (2) for this surface is This is an equation of the wth degree in c, and since at least one of the coefficients after the first is different from zero, it will have at least one root c^ + 0. On the other hand, we can find a value Cj:^0 which is not a root of this equation. Hence the surface (3) satisfies condition (2) if we let c =Cj and does not satisfy it if e= v^. But a change in the value of c merely multiplies the equation (3) by a constant and does not change the surface represented by it. Thus we see that one and the same surface can be regarded both as satisfying and as not satisfying condition (2). In other words, if ^ is non-homogeneous, (2) does not express a property of the surface (1). Assume now that ^ is homogeneous of the wth degree, and consider the class A of polynomials / whose coefficients satisfy equa- tion (2) and the class 5 whose coefficients do not satisfy this equa- tion. Our theorem will be proved if we can show that we have hereby divided the surfaces (1) into two classes, that is, that ii • This brief explanation must not be regarded as an attempt to define the concep- 'aon iiroperty, for no specific class can be defined without the use of some property. INTEGRAL RATIONAL INVARIANTS 229 Oi, 02, ■•• are the coefficients of a polynomial of class A and a'/, a!l^ •• the coefficients of a polynomial of class B, then the two surfaces /(a;, «/, z; aj , a^, •••) = 0, /(a;, y,^2; a'j', a^', •••) = 0, cannot be the same. If they were the same, the coefficients Oj, a!^,--- would be proportional to a![, a% ■ ■ ■ (cf . Theorem 7, Corollary, § 76), and therefore ^(*i> <*2' •••)= ^"^(<*p «2' "O* But this is impossible since by hypothesis <^(ai, 4, ■•■)=0, <^«, <--)^0. Thus our theorem is proved. This theorem admits of generalization in various directions. Suppose first that instead of a single surface (1) we have a system of algebraic surfaces, and that ^ is a polynomial in the coefficients of all these surfaces. Then precisely the reasoning just used shows that the equation <^ = gives a necessary and sufficient condition for a geometric property of this system of surfaces when and only when (/> is homogeneous in the coefficients of each surface taken separately. On the other hand, we may use homogeneous coordinates in writing the equations of the surfaces, and the results so far stated will obviously hold without change : Theorem 2. Let /i(a;, y,z,t; a-^, a^, ■••), A (a;, y,e,t; h^, b^, ■■■),•■■ h, a si/stem of homogeneous polynomials in {x, y, z, t) whose coefficients are aj, a^, ■•■ ; hy, h^, ■■■ ; etc. ; and let 4>{^v ^2^ ••" ' K *2' ■■■ 5 ■■■) he a polynomial in the as, Vs, etc. Then the equation <f) = expresses a necessary and sufficient condition that the system of surfaces /i = o,/2 = o,... have a geometric property when, and only when, the polynomial (j) is homogeneous in the a's alone, also in the b's alone, etc. In conclusion we note that all the results of this section can be extended at once to algebraic curves in the plane ; or, indeed, to the case of space of any number of dimensions. INTRODUCTION TO HIGHER ALGEBRA EXERCISE If, in Theorem 2, besides the surfaces /i = 0, /a = 0, ••• we also have a system "^ P°^°*' ix„ 2,1, Z„ h), (X2, 2/2, %, h), -.. and if <^ is a polynomial not merely of the a's, h'a, etc., but also of the coordinates of these points, prove that <^ = expresses a necessary and sufficient condition that this system of surfaces and points have a geometric property when and only when <^ is homogeneous in the a's alone, in the i's alone, etc., and also in (^1, yi, 2i, h) alone, in (X2, 2/2, 32, '2) alone, etc. 81. Homogeneous Invariants. From the developments of the last section it is clear that the only integral rational invariants which will be of importance in geometrical applications are those which are homogeneous in the coefficients of each of the ground-forms taken separately.* Such invariants we will speak of as homogeneous in- variants. It will be found that all the invariants which we have met so far are of this kind. An important relation between the weight and the various de- grees connected with a homogeneous invariant is given by the follow- ing theorem : Theokem 1. If we have a system of n-ary forms. (1) of degrees m-^, m^, ■ ■ • respectively, and if I(a^, a^, ■■• ; h^, b^, •■•; •••) * This statement must not be taken too literally. It is true if in the geometrical application in question we consider the variables as homogeneous coordinates and if we have to deal with the loci obtained by equating the ground-forms to zero. While this is the ordinary way in which we interpret invariants geometrically, other inter- pretations are possible. For instance, instead of interpreting the variables (x, y) as homogeneous coBrdinates on a line and equating the binary quadratic forms /i = aix' + 2 a2xy + ai^, f2=bix^ + 2b2xy + bay^ to zero, thus getting two pairs of points on a line, we may interpret (s, y) as non- homogeneous coordinates in the plape, and consider the two conies /j = 1, /a = 1. With this interpretation, the vanishing of the invariant 01O3 ~al+ bibs - bl, which is not homoeeneous in the a's alone or in the 6's alone, has a geometric meaning INTEGRAL RATIONAL INVARIANTS 231 is a homogeneous invariant of this system, of weight X, and of degree a in the a^s, y8 in the S's, etc., then (2) m^a+vi^^+ ■■■ =n\. Subjecting the forms (1) to the linear transformation (3> x^=c^^x[+ ... +c^^x'^, whose determinant we will denote by c, we get and, since by hypothesis Zis an invariant of weight \, (4) I{a[, a'^, •••; b[, h'^, •••; ■ ■ ■) = c>'I (a^, a^, ••• ; Jj, b^, •■• ; —). Every «• is a homogeneous polynomial in the c^'s of degree m^, every b' of degree m^, etc. ; and since I is itself homogeneous of degree a in the a's, ^ in the J's, etc., we see that the left-hand side of (4) is a homogeneous polynomial of degree mya + m^^+ ■••in the e^-'s. Equating this to the degree of the right-hand side of (4) in the c^-'s, which is evidently nX, our theorem is proved. An additional reason for the importance of these homogeneous invariants is that the non-homogeneous integral rational invariants can be built up from them, as is stated in the following theorem: Thbokbm 2. If an integral rational invariant I of the system (1) be written in the form _ _ ^ 1 = li + J-i+ •■■ + J-Jc where each of the I-s is a polynomial in the a's, J'«, etc., which is homogeneous in the a's alone, and also in the Vs alone, etc., and such that the sum of no two I-s has this property, then each of the functions A' -^' ■ ■ • -4 is a homogeneous invariant of the system (1). 232 INTRODUCTION TO HIGHER ALGEBRA This theorem follows immediately from the definition of an invariant. For from the identity, /^(ai, 4, ••■; 6^, 6^, ■••;•••)+••• +/i(ai, 4, ••■ ; b[, b'^, ■■■ ; ■■■) = c^[Jj(ai, asj, ■•• ; &i, ^2, ■■• ; — ) + — \-Mcv S' ■■■ ; *i' *2' "■ ' ■••)]' we infer at once the identities, Ij{a[, a'^--- ; b[, b'^, ■■■ ; •■•) = c^7i(«i, a^, ■■■ ; Jj, b^, ••• ; •••), I^.{a[, 4, ••• ; b[, b'^, ■■■ ; ■-•) = c^'Z/aj, a^, ••• ; b^, b^, ■■■ ; •••). In the case of a single w-ary form, but in that case only, we have the theorem : Theorem 3. An integral rational invariant of a single n-ary form is always homogeneous. Let /(ajj, •■•a;„; osj, a^, •.••) be the ground-form, and let / be the invariant. By Theorem 2 we may write j^ j^ + j^ + ... + j^ where Jj, ■■■ I^. are homogeneous invariants. Let the degrees of these homogeneous invariants in the a's be aj, ••• aj. respectively. Their weights are all the same as the weight of I, which we will call \. If, then, we call the degree of/, m, we have, by Theorem 1, wiotj = wX, ma^ ~ nX, ■ ■ ■ muf. = nX, from which, since m > 0, we infer That is, ii, ••• I/c are of the same degree, and / is homogeneous. Theokem 4. If we have a system of n-ary forms /j, f^, ••■ and I polynomial (j} in their coefficients, the equation <^ = gives a necessary and sufficient condition for a projective property of the system of loci in space ofn—1 dimensions, /i = 0, /2 = 0, , when, and only when, (j> is a homogeneous invariant of the system of forms f. INTEGRAL RATIONAL INVARIANTS 233 If is a homogeneous invariant, its vanishing gives a necessary and sufficient condition for a geometric property (cf. § 80), and this property must be a projective property since when we subject the loci to a non-singular collineation, ^ is merely multiplied by a non- vanishing constant. On the other hand let ^ = be a necessary and sufficient condi- tion for a projective property. In order to prove that (j> is an invariant (it must be homogeneous by § 80) let aj, a^,--- be the coefficients of f^; Jj, b^, ■■■ the coefficients of f^, etc.; and suppose that the linear transformation, ' ''ii^i + ••■ + '^im2;„, (5) carries over /j into /{ with coefficients a[, a\---\ f^ into/^ with coeffi- cients Sj, ^2' ••■ J ^tc. The polynomial (^ formed for the transformed forms is <^K,4,-; SJ, 5^, •.• ; •••), and may, since the a"s, J''s, ••• are polynomials in the a's, S's, ••■ and the c's, be itself regarded as a polynomial in the a's, J's, ••■ and the e's. Looking at it from this point of view, let us resolve it into its irreducible factors, (6) 4>{a[, 4, ••■ ; h[, V^, ■■■ ; ■■■ )=<^i(ai, a^, ■■■ ; b^, b^, ■■■ ; ••■ c^^, ■■■€„„) It is clear that at least one of the factors on the right must con- tain the c's. Let (^j be such a factor, and let us arrange it as a poly- nomial in the c's whose coefficients are polynomials in the a's, 6's, etc. Let f {a,, a„-.;b„b^,. ..;...) be one of these coefficients which is not identically zero and which is the coefficient of a term in which at least one of the c's has an expo- nent greater than zero. We can, now, give to the a's, 5's, ••• values which we will denote by ^'s, B's, ■■■ such that neither </> nor •^ vanish ; and consider a neighborhood N of the point (Aj, -4.21 ■■■ > -"l» -"2' ■■■ ' ■■■) throughout which (7) ^(aj, aj, •••; Jj, b^, •■•; •■•)=?^0i (8) -«/r(ai, a2, •••; Jj, b^, ■■■ ; ■••)^0. 234 INTRODUCTION TO HIGHER ALGEBRA Let us now restrict the a's, S's, •••to the neighborhood iVand ask ourselves under what circumstances we can have (^j = 0. If this equation is fulfilled, we see from (6) that (j> vanishes for the trans- formed loci, while, by (7), it does not vanish for the original loci. Since, by hypothesis, the vanishing of ^ gives a necessary and sufficient condition for a projective property, a transformation (5) which causes cf) to vanish when it did not vanish before must be a singular transformation. That is, if the a's, J's, ■■• are in the neigh- borhood JV, whenever (p^ vanishes the determinant c of (5) vanishes. Moreover, ^j does vanish for values of the a's, 6's, ■•• in N, for if we assign to the a's, b's, ■•• any such values, (j)j^ becomes a polynomial in the Cf/s, which, by (8), is of at least the first degree, and therefore vanishes for suitably chosen values of the e,y's. We can therefore apply the theorem for more than three variables analogous to Theorem 8, § 76, and infer that (j>^ is a factor of the determinant o ; and conse- quently, since this determinant is irreducible (Theorem 1, § 61), that (^]^ is merely a constant multiple of c. The reasoning we have just applied to (f>-^ applies equally to any of the factors on the right of (6) which are of at least the first degree in the c^/s. Accordingly (6) reduces to the form (9) ^{a[, a'^, ■•■ ; b[, b'^ •••; •••) = c*x(«i' «2' •••; K K -• ' •••)' where x i^o longer involves the c^'s. To determine this polynomial X', let us assign to the <?,y's the values 0, 1 which reduce (5) to the identical transformation. Then the a''s, 6''s, ••• reduce to the a's, 6's ••■ , while c= 1; so that from (9) we see that (^(a^, aj, ••■; Sj, Jg--- ; •■■) = xK' «2'---; ^. *j, "•; •••)• Substituting this value of x i" (9)i we see that ^ is really an in- variant. In order to avoid all misunderstanding, we state here explicitly that if we have two or more polynomials, t^j, ^^, ■■■ in the coefficients of the forms /j, the equations (^^ = ^2= ■•• =0 may be a necessary and sufficient condition for a projective property of the loci/i=0, even though ^j, ^2' ■•■ are not invariants. For instance, a necessai-v and sufficient condition that the two lines a^x-^ + a^x^ + agajj = 0, INTEGRAL RATIONAL INVARIANTS 235 coincide is the vanishing of the three two-rowed determinants of the matrix „ /, /, Cv-t C&n CCq Sj 62 Jg none of which is an invariant. Or, again, a necessary and sufficient condition that a quadric surface break up into two planes, distinct or coincident, is the vanishing of all the three-rowed determinants of its matrix, and these are not invariants. In this case we can also express the condition in question by the identical vanishing of a certain contravariant, namely, the adjoint of the quadratic form ; and this — a projective relation expressed by the identical vanish- ing of a covariant or contravariant — is typical of what we shall usually have when a single equation (^ = is not sufficient to express the condition. There are, however, cases where the condition is given by the vanishing of two or more invariants; cf. Exercise 6, §90. EXERCISES 1. Prove that if in Theorem 1 our system consists not merely of the ground- forms (1) but also of certain points and we have not an invariant / but a covariant of weight \, and of degree a in the a's, p in the 6's, etc., rj in the y'i, ^ in the z's, etc., then TOia + m2y3+ •■• = WX + 17 + ^H . 2. Extend Theorem 2 to the case of covariants. Does Theorem 3 admit of such extension ? 3. Extend Theorem 4 to the case of covariants. 4. Show that an integral rational invariant of a single binary form of odd degree must be of even degree. 5. Show that the weight of an integral rational invariant of a single binary form can never be smaller than the degree of the form. 6. Express the condition that (a) two lines, and (h) two planes coincide, in the form of the identical vanishing of a covariant or contravariant. 7. Prove that a polynomial in the coefficients of a system of n-ary forms which is homogeneous in the coefficients of each form taken by themselves, and which is unchanged when the forms are subjected to any linear transformation of determi- nant + 1, is an invariant of the system of forms. 8. Generalize Exercise 7 to the case of covariants. 238 INTRODUCTION TO HIGHER ALGEBRA 82. Resultants and Discriminants of Binary Forms. If we inter, pret (ojj, ajj) as homogeneous coordinates in one dimension, tlie equa- tions obtained by equating the two binary forms to zero represent sets of points on a line. The points given by the equation /= are the points at which the linear factors of / vanish, and the points corresponding to ^ = are the points .at which the linear factors of <;& vanish. Since two binary linear forms obviously vanish at the same point when, and only when, these linear forms are proportional, it follows that the loci of the two equations /= 0, ^ = "have a point in common when, and only when, / and ^ have a common factor other than a constant. Hence, by § 72, a necessary and suffi- cient condition that the two loci _/"= 0, ^ = have a point in common is that the resultant R of the binary forms f <^ vanish. The property of these two loci having a point in common is, however, a projective property. Thus, by Theorem 4, § 81, Theorem 1. The resultant of two binary forms is a homogeneous invariant of this pair of forms. From the determinant form of R given in § 68 it is clear that B, is of degree m in the a's and of degree n in the J's. Hence by formula (2), § 81, ^ ^ ^^_ Theorem 2. The weight of the resultant of two binary forms of degrees m and n is mn. The following geometrical problem will lead us to an important invariant of a single binary form. Let us resolve the form /, which we assume not to be identically zero, into its linear factors (cf. formula (4), § 65), f{x^, x^ = {a!lx.^-a[x^{alx^-a'^x^ ... {a'^x-^- a'^x^. The equation /= represents n distinct points provided no two of these linear factors are proportional to each other. If, however, two of these factors are proportional, we say that / has a multiple linear factor, and in this case two or more of the n points represented by the equation /= coincide. Let us inquire under what conditions this will occur. INTEGRAL RATIONAL INVARIANTS 237 Form the partial derivatives : + '^'i(<^x^-cc[x^){a'^x^-a^x^) ... {a'J^x^- a'^x^) (1) \ ^. + ■■■ +''W^i-"-'iX2)---i<-i^l-<-i^2), — = - «; {a'lz^ - a'^x^) . . . {a'ix^ - a'„x^) 2 - •-.• - a'„{a'lx^-a[x^) ••• {a'l.^x^- al_^x^). From these formulae we see that any multiple linear factor of / is a factor of both of these partial derivatives. Conversely, if these partial derivatives have a common linear factor, it must be a factor of / on account of the formula, df bx^ dx^ a formula which follows immediately from the expressions, r af — = na^xl'^ +(n- l)a^x'{--^x^ + (2) + au-ixl~\ d f But, by (1), no linear factor of / can be a factor of df/9x^ unless it is a multiple factor of/. Thus we have proved Theorem 3. A necessary and sufficient condition that f have a multiple linear factor is that the resultant of df/dx-^ and Sf/dx^ vanish. Definition. The resultant of bf/bx-^ and Sf/dx^ is called the dis- criminant off. From (2) we see that the discriminant of /may be written as a determinant of order 2 w — 2 whose elements, so far as they are not zero, are numerical multiples of the coefficients a^, a^, •■■ a„ of/. That is, this discriminant is a polynomial in the a's. Moreover, its vanishing gives a necessary and sufficient condition that the locus /= have a projective property (namely, that two points of this locus coincide). Hence, by Theorem 4, § 81, this discriminant is a * This is merely Euler's Theorem for Homogeneous Functions. 238 INTRODUCTION TO HIGHER ALGEBRA homogeneous invariant, whose degree and weight are readily deter- mined. Thus we get the theorem : Theorem 4. The discriminant of a binary form of the nth degree is a homogeneous invariant of this form of degree 2(w — 1) and of weight n(n — 1). A slight modification in the definition of the discriminant is often desirable. Let us write the binary form /, not in the above form where the coefficients are a^, a^,--a„ but, hj the introduction of binomial coefficients, in the form fix^, x^) = a^xl + Mai4-%2 + '^'^~^^ a^\-'^xl + • • • + na„_^x^3^f^ + a^xl. Then we may write i |/ = a^r' + (»* - i)«ia;r% + (^-^)(^-^) a^r^xi + n ax-, ^ ! 1 L /, r"~l ••• -f "n— 1 •''2 ) 1 A^ ^ a^xl~^ + {n- l)a,xl-^x^ + (»-l)(^-2) ^^^n-3^2 +... + a,,x^\ nox^ A ! We may then define the discriminant of / as the resultant of the two binary forms just written. We thus get for the discriminant a polynomial in the a's which differs from the discriminant as above defined only by a numerical factor, and for which Theorems 3 and 4 obviously still hold. If this last definition be applied to the case of a binary quadratic form, it will be seen that it leads us precisely to what we called the discriminant of this quadratic form in the earlier chapters of this book. EXERCISES 1. Prove that the resultant of two binary forms of degrees n and m respee- tively is irreducible. [SnGGESTioN. When 60 = 0, JJ is equal to oSq times the resultant of two binary forms of degrees n and m — \ respectively. Show that if this last resultant is irredu- cible, a is also irreducible, and use the method of induction, starting with the case n = \, m = l.] 2. Prove by the methods of this chapter that the bordered determinants of Chapter XII are invariants of weight 2. 3. The following account of Bezout's method of elimination is sometimes given : If f and <^ are polynomials In x which are both of degree n, the expression f(x)4,iy) - 4>ix)f(y) INTEGRAL RATIONAL INVARIANTS 239 vanishes, independently of y, for a value of x for which both /and ^ vanish, and is divisible hj x — y, since it is zero for x = y. Hence n^,y) ./W«(2')-«^W/(3^) x-y is a polynomial of degree (n- 1) in x which vanishes for all values of y when x is a common root of/ and 4>. Arranging F according to the powers of y, we have the expression + y<Sw + Cii» + CijX^ + ... + c^ „_ia;»-i) + /(C20+ Cn^ + £22^2 + - + C2,„-iX»-i) + + 3/"-'(c»-i,o+c»-i,ia;+ c»-i,2^'+ - +c,-,,„_ia;''-'). If this function is to vanish independently of y, the coefficient of each power of y must be zero. This gives n equations between which we can eliminate the n quantities, lx,x^, — a;"-!, obtaining the resultant in the form of the determinant. R= 'n-1,0 *'n-l, 1 ****'»— ^,! =0. With the help of the auxiliary function F we have, in this case, reduced the resultant to a determinant of the nth order, while that obtained by the method of Sylvester was of order 2 n. Criticise this treatment and makei it rigorous, applying it, in particular, to the case of homogeneous variables. 4. If /and <^ are polynomials in (x, y) of degrees n and m respectively and are relatively prime, prove that the curves /= 0, <^ = cannot have more than mn points of intersection. . [Suggestion. Show first that the coordinate axes can be turned in such a way that no two points of intersection have the same abscissa, and that the equations of the two curves are of degrees n and m respectively, after the transformation, in y alone. Then eliminate y between the two equations by Sylvester's dyalitic method. ] 5. Prove that every integral rational invariant of the binary cubic is a con- stant multiple of a power of the discriminant. [SnoGESTiON. Show that if the discriminant is not zero, every binary cubic can be reduced by a non-singular linear transformation to the normal form xf — x|. Then as in §48.] CHAPTER XVIII SYMMETRIC POLYNOMIALS 83. Fundamental Conceptions. S and S Functions. Definition 1. A polynomial F{x-^, ■■■ x,^ is said to he syvavaeXriz if it is unchanged hy any interchange, of the variables (a^j, • • • a;„). It is not necessary, however, to consider all the possible permuta- tions of the variables in order to show that a polynomial is sym- metric. If we can show that it is unchanged by the interchange of every pair of the variables, this is suificient, for any arrangement (Xa, Xi,, •■•x^) may be obtained from (x^, x^, ■■■^n) by interchanging the x's in pairs. Thus, if a^^l, interchange x^ with Xj^; then in- terchange the second letter in the arrangement thus obtained with x^; and so on. Hence we have the following theorem : Theorem 1. A necessary and sufficient condition for a poly- nomial to he symmetric is that it be unchanged by every interchange of two variables. A special class of symmetric polynomials of much importance are the 2 -functions, defined as follows: Definition 2. 2 before any term means the sum of this term and of all the similar ones obtained from it by interchanging the subscripts. Thus, for example, '2.x\ = x\-\-xl+ ■■■ + <' 2a;Ja;| = x\x\ + x\xl -f ■ • ■ + x\a^, + x-^xl + xlxl+-+x;x^ {a^fi) + + xlx{+xlxlJr-- + xlxl_^, 'S.xlxl = xlx!^ + x\xl H h x\xl It is clearly immaterial in what order the exponents a, ^8, ••• are written. Thus, ^x^a^xl — ^^i^P'r 240 SYMMETRIC POLYNOMIALS 241 If we consider any term of a symmetric polynomial, it is evident that the polynomial must contain all the terms obtained from this one by interchanging the x's. This aggregate of terms is merely a constant multiple of one of the 2's just defined. In the same way it is clear that all the other terms of the symmetric polynomial must arrange themselves in groups each of which is a constant multiple of a S. That is, Theorem 2. Every symmetric polynomial is a linear combination with constant coefficients of a certain number of 2's. Among these 2's the simplest are the sums of powers of the x's. For the sake of brevity the notation is used : S, = ^4 = x\ + xl+ ■■■ +xt (^=1, 2, ■■■). It is sometimes convenient to write S^ = n. Theoeem 3. Any symmetric polynomial in the z^s can be ex- pressed as a polynomial in a certain number of the S's. Since every symmetric polynomial is a linear combination of a certain number of 2's, in order to prove our theorem we have only to show that every 2 can be expressed as a polynomial in the S's. Now s, = xl + xt^+--+xl, S^ = x^^ + xl+ ■■■ +3^. Hence, if a + ft. From this we get the formula : (1) ^^x\ = S^S^ - S^^^ (a ^ 0). If « = /8, we have SI =xf + xf + - +3^'^ + 2x-^x-^+ 2xlxl + - = S2,+ 2-Exlx^. Hence (2) ^x;xl^l{Sl-S,:). Similarly, by multiplying 2 2:^2:^ by /S^, we get the following formulge where the three integers a, yS, 7 are supposed to be distinct: (3) * 2 xlx^^xl = S^S^Sy - S^+^Sy - S,+yS^ - S^+y S^ + 2 S^^p+^ (4) 2 xlx^xl = 1 (SlSy - S^, Sy-2 S^+y S, + 2 S^^^y), (5) 'S.xlxlxl = \{Sl - SS^^S, + 2^3^. 242 INTRODUCTION TO HIGHER ALGEBRA The proof indicated in these two special cases may be extended to the general case as follows : If we multiply together the two symmetric polynomials (6) I,xlxl-xl, S^ = ^x^^, (,k<n) we get terms of various sorts which are readily seen to be all con- tained in one or the other of the following polynomials, each of these polynomials being actually represented : Consequently, since the product of the two polynomials (6) is sym- metric, it must have the form c^l.x\*^xl -af^ + c^^xl^*^ ■:■ xt+ + c,-2xl4 - xl+^ where c^, ■•• c^+j are positive integers. Transposing, we may write 'S.xlxl ... xU^ =^\l.xl4 ... of, ■ 2a4- CiSa;»+X ... a:J — C^X^X^'' ... Xl— — Cj^liX^X^ ... 2^+'^]. Hence, if our theorem is true for ^x^ ■•• x^, it is also true for 2a;J ■•• a:^+j. But we know it is true for k = l (by definition of the /S"s), hence it is true for k=2, hence for k = S, and so on. Thus our theorem is completely proved. 84. Elementary Symmetric Functions. The notation 1.x^x^ ■■■xl may be used to represent any 2 in n variables. If yS = 7 = ■ ■ • = z/ = 0, this becomes 2a:j ov S^; if 7 =•■•. = j; = 0, it becomes 2a;Ja;S ; and so on. Let us now consider ^x^x^ •■■ ^n where «, ^8, ... v, are all or 1. The following n cases arise : a=l, ^ = 7= ... =i; = 0, Sajj, a=/3 = l, 7=:...=j; = 0, ^^iX^, '-B-ll 1-^2 The extreme case «= /8= ■•• = j/= is of no interest. We will represent these n symmetric polynomials byjOj, jSji •■■ Pm respec+'-^-ely. They are called the elementary symmetric functions. SYMMETRIC POLYNOMIALS 243 Theorem 1. Any symmetric polynomial in the x's may he ex- pressed as a polynomial in the p's. Since any symmetric polynomial in the x's may be expressed as a polynomial in the S's, it is sufficient to show that every S may be expressed as a polynomial in the ^'s. Let us introduce a new variable x and consider the polj^nomial f{x; XyX^, ■■■ Xn) = {x—X^X — X^ ■■■ (x — Xn) = X" - PjX''-^ +P23f-^ -■••+(- Ifp,. Using the factored form of/, we may write "1 r ■■■ T dx X — x-^ x — x^ x — x^ Since/ vanishes identically when x — Xi, we may write /= (a;" - 7^) - ^;(a;'-i -xr^)+-. Accordingly, _J__ = a;"-! + {Xi - jt?i)a;"-2 + {x}- p^Xi + j»,,>"-3 + • • -, X— Xi 1^ = nxf-^ + (*S'i - npi)x"-^ + {S^ - p^S^ + np^)x'^-^ + -.. OX On the other hand, we have ^ = nx'^-^ - (w - l)pix'^^ + {n- 2)p^af-s - •••. dx Hence, equating the coefficients of like powers of x in these two expressions, we have aSi-wPi = -(w-1)Pi, '^2 -Pi^i + ^P2 =(n- 2)^2' or ^1) '^-l - Pl'Sn-2 +P2Sn-3 + (" l)"-lwPn-l = ( " If-^-p {S,-p,^0, S^-PiS^ + 2p^='0, . s,-_,-piS^_^+p^s„_s- ••■ +(-i)»-i(«-iK-i=o- Now consider the identities x7-p,3^-^+Pox7-2 +(-lfp„ = (i = l, 2, ■•• n). 244 INTRODUCTION TO HIGHER ALGEBRA Multiplying these identities by ajj"", •■■ a;^"" respectively and adding the results, we have (2) S,-p^8,_^+p^S,_^--+{-lfpA-n=0 {k=n,n+l,-). Formulae (1) and (2) are known as Newton's Formuloe. By means of them we can compute in succession the values of Sy, S^, •■■ as polynomials in the ^'s : S,=pl-2p„ (3) Thus our theorem is proved. It will be noted that Newton's formulae (1) cannot be obtained from (2) by giving to k values less, than n. The necessity for two different sets of formulse may, however, be avoided by introducing the notation «, = j, = . . . = n Fn+l — Pn-l-2 — — "• Then all of Newton's formulse may be included in the following form: (4) S,-p,S,^, + -+{-lf-^p,_,S,+{-lfkp,=0 (^=1,2,-). Using this notation, we see that the explicit formulae (3) for expressing the <S"s in terms of the p's are wholly independent of the number n of the x's. Since the formulae referred to in the last section for expressing the S's in terms of the <S"s are also independent of n, we have established Theorem 2. J^we introduce the notation p„^^=p„^^= ••■ =0 and use JVewton's Formuloe in the form (4), the formula for expressing any ^ as a polynomial in the p^s is independent of the number n of the x's. When we have k polynomials in n variables we say that there exists a rational relation between them when, and only when, a polynomial in h variables exists which is not identically zero, but which becomes identically zero as a polynomial in the a;'s when each s is replaced by the corresponding/, J'(/i, •••/^)=0. SYMMETRIC POLYNomiAT^S 245 Theorem 3. There exists no rational relation between the elemen- tary symmetric functions in n variables p.^., ■■■Pn- For let J'(si, ••• s„) be any polynomial in n variables which is not identically zero, and let (a^, •■■ a„) be a point at which this polynomial does not vanish. Determine (x^, ■■■ x„) as the roots of the equation x" - Oia;"-! + a^x"-^ _...+(_ lya^ = 0. For these values of the x's, the jo's have the values a^, •■•«„, and. therefore -P(^i, ■■■Pn) does not vanish for these x's, and is con- sequently not identically zero as a polynomial in the x^s. Thus our theorem is proved. Corollary. There is only one way in which a symmetric poly- nomial in (x-^, ■ ■ ■ x^) can be expressed as a polynomial in the elementary symmetric functions p-^, ■■■ p„. For if/ is a symmetric polynomial, and if we had two expressions ^*'''*' f(x„-x^)^cl,,(p„-p,), f(x-^,--x„) = 4>^(p^,---p^), then by subtracting these identities from one another we should hav« as an identity in the x's, 't'liPv ■■■Pn)-^i{Pv ■■■Pu)^^- This, however, would give us a rational relation between the ^'s. unless 4,iz„-z:)^uh^...z:). Thus we see that the two expressions for /are really the same. EXERCISES 1. Obtain the expressions for the following symmetric polynomials in terms of the elementary symmetric functions : 2. Prove that every symmetric polynomial in (xi,--Xn) can be expressed in one, and only one, way as a polynomial in Si, ■■■ S„. 85. The Weights and Degrees of Symmetric Polynomials. We will attach to each of the elemeutary symmetric functions pt a weight equal to its subscript, cf. § 79. Theorem 1. A homogeneous sy/nmetric polynomial of degree m in the x's, when expressed in terms of the p's, is isobaric of weight m. 246 INTRODUCTION TO IIKiHER ALGEBRA Let (1) fi^V % • ■ ■ ^n) = <^( Pv PV — Pn) be such a polynomial. Since p^ is a homogeneous polynomial of the first degree in the rr's, p^ of the second, etc., any term of <^, when written in the x's, must be a homogeneous polynomial of degree equal to the original weight of the term. Thus, for example, the term 6 pIp2Pz whose weight is 13, when written in the x's will be a homo- geneous polynomial of degree 13. Accordingly an isobaric group of terms when expressed in terms of the x's will, since by Theorem 3, § 84, it cannot reduce identically to zero, be homogeneous of the same degree as its original weight. If then (f) were not isobaric, /would not be. homogeneous, and our theorem is proved. Corollary. If f is non-homogeneous and of the mth degree, <j) is nonrisoharic and of weight m. Theorem 2. A symmetric polynomial in (x^, ■■■ «„), when written in terms of the elementary symmetric functions p^, ■ ■ ■ p^i i^iU ie of the same degree in the p's as it was at first in any one of the x^s. Let/ be the symmetric polynomial, and write f{Xv a^g, ••• Xn)^<^{Pv> Pv •■■ Pn\ and suppose /is of degree m in x-^ (and therefore, on account of the symmetry, in any one of the 2;'s), and that is of degree fi in the ^'s. We wish to prove that m = jj,. Since the p's are of the first degree in Xy, it is clear that m</x.. If (\) is non-homogeneous, we can break it up into the sum of a number of homogeneous polynomials by grouping together all the terms of like degree. Each of these homogeneous polynomials in the p's can be expressed (by substituting for the p's their values in terms of the a;'s) as a symmetric polynomial in the a;'s. If our theo- rem were established in the case in which the polynomial in the ^'s is homogeneous, its truth in the general case would then follow at once. Let us then assume that ^ is a homogeneous polynomial. The theorem is obviously true when n=l, since then p.^= — x^ It will therefore be completely proved by the method of mathematical in- duction if, assuming it to hold when the number of x's is 1, 2, ••• w— 1, we ean prove that it holds when the number of a;'s is n. SYMMETRIC POLYNOMIALS 247 For this purpose let us first assume that p„ is not a factor of every term of (f). Then <}>{pi, ■■■ Pn-v ^) ^^ ^'^^ identically zero but is still a homogeneous polynomial of degree yit in (p-^, ••■ Pn-i)' Now let Xn = 0. This makes p„ = 0, and gives the identity (2) /(^i, -a^^-i, 0)^<^(p[...K-i,0), where p'l, ••• p'n-i a-re the elementary symmetric functions of {x-^, ■•■ a;„_i), and f{xp •■• x„_.^, 0) is a symmetric polynomial of degree m-^ in x-^, where m-^-^m. From the assumption that our theorem holds when the number of x's is w— 1, we infer from (2) that /i = »Wj^ ^ m ; and since we saw above that /j, cannot be less than m, we infer that (i, = m, as was to be proved. There remains merely the case to be considered in which p„ is a factor of every term of <^. Let p* be the highest power of _p„ which occurs as a factor in <p. Then 4>{PV ■■■ Pn)=Pn4>l{Pv -Pn), where ^^ is a polynomial of degree /i — h Putting in for the ^'s their values in terms of the x's, we get (3) /(a^i, ■■■ x„) = x^xl • ■ • a;*/i(a;i, — x„), where (4) /i (o^i, ■ ■ • a;„) = ^1 (;?i, • • • p„). From (3) we see that /j is of degree m — kin x^, and from (4), since </)j does not contain jo„ as a factor, that the degrees of /j in x-^, and of (f)i in the »'s are equal, m — K = fn— K. From which we see that m = fi, a.s was to be proved. The two theorems of this section are not only of theoretical importance, they may also be put to the direct practical use of facilitating the computation of the values of symmetric polynomials in terms of the p's. In order to illustrate this, let us consider the symmetric function f{x^, ■■■ x^) = ^xlx^Xy Since / is homogeneous of the fourth degree in the a:'s, it will, by Theorem 1, be isobaric of weight 4 in the ^'s. Since it is of the 248 INTRODUCTION TO HIGHER ALGEBRA second degree in x-^, it will, by Theorem 2, be of the second degree in them's. Hence (5) Sarf^aa-g = Ap^p^ + Bpl + Qp^, where A, B, and Q are independent of the number n (Theorem 2, § 84), and may be determined by the ordinary method of unde- termined coefficients. Take w = 3, so that p^ = 0. Letting x-^^ = 0, ^2= ajg = 1, we have jBj = 2,jt>2 = 1,^3 = 0. Substituting these values in (5), we find B=0. Letting x^= —1, x^^x^^ 1, we have p-y = l, p2= —^, P3= — 1. which gives A = l. Now let w = 4, Xi = X2 = xg = x^ = l. From this we fipd p-^ = i, p^ = 6, pg = i, p^ = 1. Substituting this in (5) gives 0= —i. Hence ^x^x^x^ = p-^ps - 4p^. EXERCISES 1. The symmetric function /(Xl, ••• Xn)='S,xlX2Xs + 'SiX^Xl + 'SiXlXiXsXi is homogeneous of the fourth degree in the x's, and is of the second degree in xi ; hence, when written in terms of the^j's, it will have the same form, Apip^ + Bpl + Cpi, as the above example. Compute the values ot A, B, and C. 2. If /(xi, X2, xs) s (xi—X2y(xi — 13)^^2 — xi)^, show that f(xi, X2, Xs) = - 27 pI - ipl + l^pipipz - ^pIp3 ^-pipl- 86. The Resultant and the Discriminant of Two Polynomials in One Variable. Let f{x) = a;" + a^a;"-! + a^x""-"^ + ■ ■ • + a„ = {x- a-J^x - a^) ■■■ {x- «„), ^{x) = x^ + h^x-^-^ + h^x'^-'^ + ■■■ +h^ = {x-fi,){x-l3^)-{x-^^), be two polynomials in x, and consider the product of the mn factors (1) («l-/3l)(«l-^2)-(«l-^m) .(««-- /3l)(«» - ^2) ■■■ (on-^m)- SYMMETRIC POLYNOMIALS 249 Phis product vanishes when, and only when, at least one of the «'s is equal to one of the yS's. Its vanishing therefore gives a necessary and sufficient condition that /and ^ have a common factor. More- over, the product (1), being a symmetrical polynomial in the '«'s and also in the jS's, can be expressed as a polynomial in the elementary symmetric functions of the «'s and /S's, and therefore as a polynomial in the a's and J's. This will be still more evident if we notice that the product (1) may be written ^(«i)<^(«2) ••• ^K> In this form it is a symmetric polynomial in the «'s whose co- efficients are polynomials in the 6's, and it remains merely to bring in the a's in place of the «'s. We thus see that the product (1) may be expressed as a poly- nomial F^a^, ■■■ a„; 6j, ■•• b^) in the a's and b's whose vanishing gives a necessary and sufficient condition that / and ^ have a common factor. In § 68 we also found a polynomial in the a's and J's whose vanishing gives a necessary and sufficient condition that / and ^ have a common factor, namely the resultant M of/ and (j>. We will now identify these two polynomials by means of the following theorem : Theoeem 1. The product (1) differs from the resultant B of f and <f) only hy a constant factor, and the resultant is an irreducible polynomial in the a's and b's. In order to prove this theorem we will first show that this prod- uct (1), which we will call F(a^, ■■•«„; 5^, ••• 5„), is irreducible. This may be done as follows : Suppose F is reducible, and let Fia^, ••• a„ ; b^, - b^)~F^(a^, ■■■ a„; b^, ■•■ J„) F^ia^, ••• a„ ; Jj, ••• JJ, where F^ and F^ are polynomials neither of which is a constant. Then, since the a's and b's are symmetric polynomials in the a's and (8's, J'l and F^ may be expressed as symmetric polynomials i^j and <f>, in the a's and ^'s, and we may write ■(«l-y8i)(«i-^2)-"(«l-/3™) («2 - /3i) («2 - ySj) - («2 - /8™) .(««-/8i)(a„-^2)-K-/3«)- 250 INTRODUCTION TO HIGHER ALGEBRA The factors on the right-hand side of this identity being irreducible, we see that 0j must be composed of some of these binomial factors and ^2 of ''he others. This, however, is impossible, since neither ^j nor <^2 would be symmetric. Hence F is irreducible. Now, since jP=0 is a necessary and sufficient condition ior f{x) and ^x) to have a common factor, and i? = is the same, any set of values of the a's and 6's which make F= will also make R=Q. Hence by the theorem for n + m variables analogous to Theorem 7, § 76, F is a factor of R. Also, since F is & symmetric polynomial in the a's and /3's of degree m in each of the a's and n in each of the /S's, by Theorem 2, § 85, it must be of degree m in the a's and n in the 5's. But R is of degree not greater than m in the a's and n in the J's, as is at once obvious from a glance at the determinant in § 68. Hence F, being a factor of R, and of degree not lower than R, can differ from it only by a constant factor. Thus our theorem is proved. Let us turn now to the question: Under what conditions does the polynomial fix) have a multiple linear factor ? Using the same notation as above, we see that the vanishing of the product («i - «2)(«i -"a) ■••(«!-«») " (a2-«3)"-("2-«") («»-!—««). = ^(«i» ••■«») is a necessary and sufficient condition for this. P is not symmetric in the a's, since an interchange of two subscripts changes P into — P. If, however, we consider P^ in place of P, we have a symmetric polynominal, which can therefore be expressed as a polynomial in the a's, C^K •■•«.)? =^K •••«.)• Moreover, ^= is also a necessary and sufficient condition that/(ir) have a multiple linear factor. On the other hand, it is easily seen that f{x) has a multiple linear factor when and only when f{x) and f'{x) have a common linear factor. A necessary and sufficient condition for f{p) to have a mul- tiple linear factor is therefore the vanishing of the resultant of /(~^, and f\x). This resultant we will call the discriminant A of fix). It is obviously a polynomial in the coefficients of /. SYMMETRIC POLYNOMIALS 251 Theorem 2. The polynomials F and A differ only hy a constant factor, and are irreducible. The proof of this theorem is similar to the proof of Theorem 1, and is left to the reader, EXERCISES 1. Compute by the use of symmetric functions the product (1) for the two polynomials x^ + a,x + a^, x^ + bix + 62, ttud compare the result with the resultant obtained in determinant form. 2. Verify Theorem 2 by comparing the result of Exercise 2, § 85, with the discriminant in determinant form of the polynomial a^ + Oix^ + OiX + Og. CHAPTER XIX POLYNOMIALS SYMMETRIC IN PAIRS OF VARIABLES 87. Fundamental Conceptions. 2 and 5 Functions. The variables (ajj, ••■ a;„) which we used in the last chapter may be regarded, if we wish, not as the coordinates of a point in space of n dimensions, but rather as the coordinates of n points on a line. In fact this is the interpretation which is naturally suggested to us by the ordinary applications of the theory of symmetric functions (cf. §86). Looked at from this point of view, it is natural to generalize the conception of symmetric functions by considering n points in a plane, (1) {^vyx)A^vy%)-^-^-^n-,yn)- Definition. A polynomial, in the coordinates of the points (l)is said to he a symmetric polynomial in these pairs of variables if it is imchanged by every interchange of these pairs of variables. As in the case of points on a line, we see that it is not necessary to consider all the possible permutations of the subscripts in order to show that a polynomial F is symmetric. It is sufficient to show that F is unchanged by the interchange of every pair of the points (1). We will introduce the 2 notation here precisely as in the case of single variables. Thus, for example, 2 a;J't/f • = a;J'2/?> + 4>«/|> + • • • + xl^yi\ 2 xlY^^xl^yl' = xl'yl^x^^Y^' + a^J-yf 'a;»^«/f ^ + • • ., and so on. As in the case of single variables, it is clear that the order in which the pairs of exponents "j, /S^; e^^' ^^2' ••• ^^^ written is imma- terial ; and also that every symmetric polynomial in the pairs of variables (1) is a linear combination of a certain number of 2's. 252 POLYNOMIALS SYMMETRIC IN PAIRS OF VARIABLES 253 We introduce the notation S,, s Sa^^j ^ xly{ + o^yl + ... + ^„y'„ ^^ = ^^ J| ;- j. Theorem. J.wt/ symmetria polynomial F(x^, y^\ ••• a;„, y„) ma?) 5e expressed as a polynomial in these S^s. The proof of this theorem is exactly like that of Theorem 3, § 88, and is left to the reader. 88. Elementary Symmetric Functions of Pairs of Variables. Every S function of n pairs of variables may, by giving to the «'s and ;8's suitable values, be written in the form (1) 'Exl^yl'x^^yl;' ■■■ x^'-y^". Definition. The function (1) is said to he an elementary sym- metric function of the pairs of variables {x^, y^), ■ ■ ■ (a;„, «/„) when, and only when, ^ r, -i -• ■. « v hut not all the as and yS's are zero. We shall adopt the following notation for these elementary sym. )ric functions : ^^^^2^„ Po^^^ ^v fn.Q'=^-^'2,'"^m ' ■ ■ Pi,n-i — ^2;^ • • • 2;^^^.! • • • Z/„, • • • p^n — ^1^2 '" Vn- It is clear that there are a finite number, ^ n (n+ 3), of Pi/s, but an infinite number of Sf/s. We will attach to each|? a weight with regard to thex^s equal to its first subscript and a weight with regard to the ys equal to its second subscript. When we speak simply of the weight of jt?,;, we will mean its total weight, that is, the sum of its subscripts. Theorem. Any symmetric polynomial T'ix^, y-^', • ■ • a^n, y^ may he expressed as a polynomial in the pijS. Since, by the theorem in § 87, any such polynomial may be expressed as a polynomial in the SyS, it is sufficient to show that the S,j9, may be expressed as polynomials in the p^/s. 254 INTRODUCTION TO HIGHER ALGEBRA and form the elementary symmetric functions of these ^'s : ■^1 - ^^1 = '^^^i + l^'^Vi - '^Pw + f^Pov = ^¥20+ Vi'n + ^Vo2' "^3 - ^ ?1 ?2 ?3 = ~^^PZ0 + '^Vi'ai + ^/^Vl2 + /^¥o3' '^n = hh-L = ^"Pno + ^"''PP^l, 1 + ^"-^ M'^Pn-Z, 2+-+ /*"^o». Also let o-asS^* (/fc=l,2, •••) Let a and he positive integers, or zero, but not both zero. Then o-.+p = 2 ^»+^ = \-'+^l.x'^+^ + X''+^- V2a;J+P-iz/j + -.. But by Theorem 1, § 84, we may write where J' is a polynomial. Hence where '^ is a polynomial. Regarding this as an identity in (\, fi) and equating the coefficients of the terms containing V/^^, we get an identity in the x's and y's, Sa?^^(Pio,-Pon), where ^ is a polynomial in the p's. Thus our theorem is proved. Theorem 3, § 84, does not hold in the case of pairs of variables, as relations between the ^n(n+3) pys do exist; for example, if n=2, the polynomial 4^20^02 -^^20^01 -^^10^02 + PloPuPoi - Pll vanishes identically when the p's are replaced by their values in terms of the x's. It does not vanish identically when w = 3. In view of the remark just made, it is clear that the represen- tations of polynomials in pairs of variables in terms of the pi/s will not be unique. For further information concerning the subjects treated in this section, the reader may consult Netto's Algebra, Vol. 2, p. 63. POLYNOMIALS SYMMETRIC IN PAIRS OF VARIABLES 255 EXERCISES 1. Prove that a polynomial symmetric in the pairs of variables -(Xi, y^ and which is homogeneous in the x's alone of degree n and in the ^'s alone of degree m can be expressed as a polynomial in the jj^'s isobaric of weight n with regard to the x'i, and m with regard to the y's. 2. Express the symmetric polynomial S xly^ya in terms of the ^,;;'s by the method of undetermined coefficients, making use of Exercise 1. 3. A polynomial in (xi, yi, zi; xs, yi, 22; ••• Xn, y„, z») which is unchanged by every interchange of the subscripts is called a symmetric polynomial in the n points {xi, yi, zi). Extend the results of this section and the last to polynomials of this sort. 89. Binary Symmetric Functions. The pairs of variables (x^, y^, • •• (a;„, «/„) may be regarded as the homogeneous coordinates of n points on a line as well as the non-homogeneous coordinates of n points in a plane. It will then be natural to consider only sym- metric polynomials which are homogeneous in each pair of variables alone. Such polynomials we will call Unary symmetric functions. Most ©f the j»,/s of the last section are thus excluded. The last w+1 of them (p^o'i'"-!,!' ■■" Po»)' tiowever, are homogeneous of the first degree in each pair of variables alone. We will call them the elementary Unary symmetric functions. Theorem 1. Any Unary symmetric function in {x-^,yi\ ■•■ ^n-^Vn) can he expressed as a polynomial in {p„o,Pn~i,v '" Po")- If we break up our binary symmetric function into 2's, it is clear that each of these 2's will itself be a binary symmetric function, or, as we will say for brevity, a binary 2. It is therefore sufficient to prove that our theorem is true for every binary 2. The general binary 2 may be written 2 x^y{'af^yl' ■ ■ ■ K'^fn" («i ^ «2 ^ • • • ^ ««> where, if we denote by m the degree of this 2 in any one of the pairs of variables, m ■■ ^ «i + /3i = «2 + ^2 =••■=«« + '^«- Let U8 assume for the moment that none of the y's are zero, and let 256 INTRODUCTION TO HIGHER ALGEBRA Now consider the elementary symmetric functions of these X's: Pan Pon P — X X ■■■ X —P«o We may write Po^ (1) ^--i-^^2-yi'-^^^" ^sx^.x^. ... X- = *(A, ... P„), where, since we have assumed «i > a2 = • • ■ = "ni •!* is a polynomial of degree a^ in the P's (Theorem 2, § 85). Hence we may write (2\ ^(jP ■■■ p w 'P(.i'o»''/'i,"-i' ••• j^«o)' ^ where ^ is a homogeneous polynomial of degree k^ We thus get from (1) and (2) (3) 2 xlY,^ ■ ■ ■ x:yJr = pfc</>(^o»' Pi, n-v • ■ ■ Pno)^ an equation which holds except when one of the «/'s is zero. Since each side of (3) can be regarded as a polynomial in the x's and ys, we infer, by Theorem 5, § 2, that this is an identity, and our theorem is proved. By Theorem 1, § 85, <I> is isobaric of weight a^-i-a2+ ■■■ + a„ in the P's. Hence 2a;J'«/J' .■• a;°"«/^", when expressed in terms of these (n + l)pij'^, is isobaric of weight aj + «2 + ••• + «„, provided we count the weight of the p^jS with regard to the x's. Passing back now to an aggregate of a number of such 2's, we get Theorem 2. If a binary symmetric function is homogeneous in the n x's {or y's) of degree k, it will, when expressed in terms of PnQi Pn-hv ■■■ Pony ^^ isobaric of weight k with regard to the x's {or y^s). We have seen in the proof of Theorem 1 that the polynomial <^ in (3) is a homogeneous polynomial of degree a^ in the ^'s ; so that ^^i'"^//' ... a;^ «//" is a homogeneous polynomial of degree a^ + ySj = m in the ^'s. Hence Theorem 3. Any binary symmetric function of degree m in each pair of variables will, when written in terms of pnQ,p„_i^i, ■■■ Pf^n ^^ "^ homogeneous polynomial of degree m in these p's. POLYNOMIALS SYMMETRIC IN PAIRS OF VARIABLES 257 EXERCISES 1. Prove that no rational relation exists between pno, ••• pom and hence that a binary symmetric function can be expressed as a polynomial in them in only one way. 2. By a ternary symmetric function is meant a symmetric polynomial in n points (xi, T/i, Zj) -which is homogeneous in the coordinates' of each point. Extend the results of this section to ternary symmetric functions. Cf. Exer- cise 3, § 88. 90. Resultants and Discriminants of Binary Forms. It is the object of the present section to show how the subject of the re- sultants and discriminants of binary forms may be approached from the point of view of symmetric functions. ^®* f{x^, x^ = a^x\ + a^xl-^x^ + ■■■ + a^x^ = (a'^Xj^ — a[xr^){a'^x^ — a'^x^) ■■■ (a'^j — w^a^z)' 4>{wi, x^ = b,xf + 6ia;f-%2 + •■■ + b^x^ s (/31'xi - ^[xj^^'^x^ - ^'^^) ••• (,S;>i - /3W. be two binary forms. Each of these polynomials has here been written first in the unfactored and secondly in the factored form. By a comparison of these two forms we see at once that the elemen- tary binary symmetric fractions of the n points are and of the m points (/3[, /S'Di^'^, ^'^), ■ ■ ■ {0L 0L) are ' Sa -o'-*i'^2'-(-ir*« Let us now consider the two linear factors a'!x.^^-a'iX^, 0!x^- ft'^x^. A necessary and sufficient condition for these factors to be propor- tional is that the determinant vanish. Let us form the product of all such determinants : P = ia'lP'^ - a\^'L) {a'<0^ - a',0i) -{a'lff^ - a' A) J 258 INTRODUCTION TO HIGHER ALGEBRA The vanishing of this product is a necessary and sufficient con- dition that at least one of the linear factors of / be proportional to one of the linear factors of 0, that is, that /and (f> have a common factor which is not a constant. We may obviously reduce J-* to the simple form P ^/(ySi, /3i')/(/3^, 13'^) -/(/3L, 131). In this form it appears as a homogeneous polynomial of the mth degree in the as, and as a symmetric polynomial in the m points (/3J, j8f ). Moreover, it is obviously a binary symmetric function which is of the nth degree in the coordinates of each point. Consequently, by Theorem 3, § 89, it can be expressed as a homogeneous polynomial of the wth degree in the elementary binary symmetric functions of the points (/S^, /3"), that is, in the h's. Thus we have shown that the product P can he expressed as a polynomial in the a's and 5's which is homogeneous in the a's of degree m and in the Vs of degree n. In § 72 we found another polynomial in the a's and J's, whose vanishing also gives a necessary and sufficient condition for/ and <j) to have a common factor, namely, the resultant R. We will now identify these polynomials by means of the following theorem : Theorem 1. The product P differs from the resultant P of f and <j) only hy a constant factor, and the resultant is an irreducible poly- nomial in the a's and Fs. We may show, in exactly the same way as in the proof or heo rem 1, § 86, that P, when expressed as a polynomial in the a'- and 6's, is irreducible. Since P = and i? = each give a necessary and sufficient condition for /and <^ to have a common factor, any ^et of values of the a's and J's which make P = will also make if = 0. Thus by Theorem 7, § 76, P is a factor of P. We have seen that P is of degree m in the a's and n in the J's. The same is also true of P, as may easily be seen by inspection of the determinant of § 68. Hence, P being a factor of P, and of the same degree, can differ from it only by a constant factor. Thus our theorem is proved. Let us now inquire under what conditions the binary form f(x-^, x^) has a multiple linear ' factor. Using the same notation as above, we see that the vanishing of the product «4 - «i<)(« - aiO - (« - «ia;0- («»-!«» -<-l«!I). = PiK,<; -<,«^') POLYNOMIALS SYMMETRIC IN PAIRS OF VARIABLES 25& is a necessary and sufficient condition for this. P^ is not symmetric in tlie pairs of as, since an interchange of two subscripts changes P. into -Pj. If, however, we consider P\ instead of P^, we have a binary symmetric function which can be expressed as a polynomial in the a's Moreover, P vanishes when, and only when, Pj does. Accordingly P= is a necessary and sufficient ■ condition for f(x^, x^) to have a multiple linear factor. But the vanishing of the discriminant A (cf. § 82) of /(ajj, x ) is also a necessary and sufficient condition for this. Theorem -2. F and A differ only hy a constayit factor, and are irreducible. The proof of this theorem, which is practically the same as that of Theorem 1. is left to the reader. If we subject the two binary forms / and 0, which we may sup- pose written in the factored form, to the linear transformation [ 2^2 ~ ^21^1 "I" '^22'''2' we get two new binary forms iA'i^[ - A[x',){A'ix[ - A',x',) ... (A':x[ - A'„4), {B'ix[ - B[x',){B'lx\ - B'^x-) -{B'y, - B'J^), where ^f= «,% - «,V ^= ^"<'n-^hv A'i=- «f 6-12 + "5^22, Pi = - /3;'ei2 + ^l-<f-^, so that A'JB'j - A[B'! = c{a'^^'j - a'S'j), where c is the determinant of the transformation (IV Since the linear transformation (1) may be regarded as carrying over the « 's and yS's into the J.'s and P's, the last written identity shows us that «,"/3y — «j'/3" is, in a certain sense, an invariant of weight 1. It can, however, not be expressed rationally in terms of the a's and Vs. Such an expression is called an irrational invariant Since the resultant of / and ^ is the product of mn such irra- tional invariants of weight 1, it is evident that the resultant itself i6 an invariant of weight mn. Thus we get a new proof of this fact, independent of the proof given in § 82. A similar proof can be used in the case of the discriminant of a binary form. 260 INTRODUCTION TO HIGHER ALGEBRA EXERCISES Develop the theory of the invariants of the binary biquadratic f(x^, x^) =a^\ + i: a^x{x^ + 6 a^\xl + 4 a^^xl + a^x* = (a'/ii - a.[x^) (ct^'xj - a'^^ {a'^x^ - a'^^) {a^'x^ - 06^3:,) along the following lines : 1. Start from the irrational invariants of weight 2, A = (al'a^ - a;0(a^'a{ - c^O, c =(«;'<-«;<) K«3 - «2«3'). whose sum is zero, and the negatives of whose ratios are the cross-ratios, of the four points « <), (< O, ("s. "a'). («4- «")• 2. Form the further irrational invariants of weight 2 £i = B_C, li:2 = C-A, Es=A~B; and prove that every homogeneous symmetric polynomial in Ei, Ei, Es is a binary symmetric function of the four points («■, «['), and therefore an integral rational invariant oif. 3. In particular G2=EiE2 + E^Es + EsEij G^^EiE^E^ are homogeneous integral rational invariants of weights 4 and 6, and of degrees 2 and 3 respectively. Prove that G2= - 36^2, Gj>=iB2ga, where ^2=0004— 4aia3+ 3 Oj, ^3=000204 + 2 010203— aoal — alai — a|. These expressions g^ and ^s are the simplest invariants of/.* 4. Prove that the discriminant A of /is given by the formula ^=gl- 27 <f,. 5. If A ^ 0, prove that jra = is a necessary and sufficient condition that the four points f= form a harmonic range ; and that g2 = is a necessary ana sufficient condition that they form an equianharmonic range. (Cf. Exercise ;! § 33.) 6 Prove that (72 = g'3 = is a necessary and sufficient condition that / have at least a threefold linear faotor.f * They are among the oldest examples of invariants, having been found by Cayley and Boole in 1845. t Notice that we here have a projective property of the locus / = expressed by the vanishing of two integral rational invariants; cf. the closing paragraph of § 81 POLYNOMIALS SYMMETRIC IN PAIRS OF VARIABLES 261 V. If A is the absolute irrational invariant i.e. one of the cross-ratios of the points /=0, prove that the absolute rational invariant j /= — can be expressed in the form 27 (A,-l)2\2 • 8. Prove that a necessary and sufficient condition for the equivalence of two biquadratic binary forms neither of whose discriminants is zero is that the inva- riant / have the same value for the two forms. 9. Prove that a necessary and sufficient condition ifor the equivalence with regard to linear transformations with determinant»+ 1 of two biquadratic binary forms for which ^2 and ga are both different from zero is that the values of g^ and ga be the same for one form as for the other. 10. Prove that if the discriminant of a biquadratic binary form is not zero, the form can be reduced by means of a linear transformation of determinant + 1 to the normal form ^ 3 34 4 3^X2 — g2XiX2 — gsxl 11. Prove that every integral rational invariant of a biquadratic binary form is a polynomial in g^ and gs. 12. Develop the theory of the invariants of a pair of binary quadratic forms along the same lines as those just sketched for a single biquadratic form. 13. Prove that every integral rational invariant of a pair of quadratic forms in n variables is an integral rational function of the invariants ©0, ••• ®„ of § 57. [Suggestion. Show first that, provided a certain integral rational function of the coefficients of the quadratic form does not vanish, there exists a linear transfor- mation of determinant + 1 which reduces the pair of forms to ai3^ + a2xl+ ■■■ -Va-A Pl4 + P24+ - +PnXl- Then show that every integral rational invariant of the pair of quadratic forms can be expressed as a binary symmetric function of (cci, /Si), (ota, ft), .•■ (a,, /3„), and that the ©'s are precisely the elementary binary synunetrio functions.] CHAPTER XX ELEMENTARY DIVISORS AND THE EQUIVALENCE OF X-MATRICES 91. X-Matrices and their Elementary Transformations. The theory of elementary divisors, invented by Sylvester, H. J. S. Smith, and, mor^ particularly, Weierstrass, and perfected in important respects by Kronecker, Frobenius,«and others, has, in the form in which we will present it,* for its immediate purpose the study of matrices (which without loss of generality we assume to be square) whose elements are polynomials in a single variable X. Such matrices we will call X-matrices.f The determinant of a X-matrix is a polynomial in X, and if this determinant vanishes identically, we will call the matrix a singular X-matrix. By the rank of a X-matrix we under- stand the order of the largest determinant of the matrix which is not identically zero. We have occasion here, as in § 19, to consider certain elementary transformations which we define as follows: Dbfinitiox 1. By an elementary transformation of a \-matrix we understand, a transformation of any one of the following forms : (a) The interchange of two rows or of tivo columns. (b) The multiplication of each element of a row (or of a column) by the same constant not zero. (e) The addition to the elements of a row (or column) of the products of the corresponding elements of another row {or column) by one and the same polynomial in X. * Various modifications of the point of view liere adopted are possible and im- portant. First, we may consider matrices wliose elements are polynomials in any num- ber of variables. Secondly, we may confine ourselves to polynomials whose coefficients lie in a certain domain of rationality. Thirdly, we may approach the subject from the side of the theory of numbers, assuming that the coefficients of the polynomials are integers. The simplest case here would be that in which the elements of the matrix are themselves integers ; see Exercise 2, §91, Exercise 3, §92, and Exercise 2, §94. t The matrix of a pencil of quadratic forms is an important example of a X-matriz to which the general theory will be applied in Chapter XXII. 262 ELEMENTARY DIVISOKS AND X-MATRICES 263 If we pass from a first matrix to a second by an elementary trans- formation, it is clear that we can pass back from the second to the first by an elementary transformation. Thus the following defini- tion is justified: Dbpinitioit 2. Two X-matrices are said to he equivalent if it is possible to pass from one to the other hy means of a finite number of elementary transformations. We see here that all X-matrices equivalent to a given matrix are equivalent to each other; and, as in §19, that two equivalent X-ma- trices always have the same rank. The rank of a \-matrix is nOt, however, the only thing which is left unchanged by every elementary transformation. In order to show this we begin with Lemma 1. If the polynomial <f>(\) is a factor of all the i-rowed determinants of a X-matrix a, it will be a factor of all the i-rowed determinants of every X-matrix obtained from a by means of an elemen- tary transformation. If the transformation is of the type (a) or (5) of Definition 1, this lemma is obviously true, since these transformations have no effect on the i-rowed determinants of a except to multiply them by con- stants which are not zero. If it is of the type (e), let us suppose it consists in adding to the elements of the pth column of a the corre- sponding elements of the qth. column, each multiplied by the poly- nomial -v/r (X). Any i-rowed determinant of a which either does not involve the pth. column, or involves both the pth and the g'th, will be unaffected by this transformation. An i-v^wed determinant which involves the ^th column but not the qth. may be written after the transformation in the form A ± -f (X)5, where A and £ are i-rowed determinants of a; so that here also our lemma is true. Theokem 1. If Si and b are equivalent X-matrices of rank r, and DlX) is the greatest common divisor of the i-rowed determinants {i<r) of a, then it is also the greatest common divisor of the i-rowed determi- nants ofh. For by our lemma, D^X) is a factor of all the i-rowed determi- nants of b ; and if these determinants had a common factor of higher degree, this factor would, by our lemma, be a factor of all the i-rowed determinants of a; which is contrary to hypothesis. 264 INTRODUCTION TO HIGHER ALGEBRA The theorem just proved shows that the greatest common divisors 2)j(X,), ••■ D^(\) are invariants with regard to elementary transfor- mations, or, more generally, that they are invariants with regard to all transformations which can be built up from a finite number of elementary transformations. In point of fact they form, along with the rank r, a complete system of invariants. To prove this we now proceed to show how, by means of elementary transformations, a X- matrix may be reduced to a very simple normal form. Lemma 2. If the first element * f{X) of a\-matrix is not identically zero and is not a factor of all the other elements, then an equivalent matrix can he formed whose first element is not identically zero and is of lower degree thanf. Suppose first there is an element /^(A.) in the first row which is not divisible by/(X.) and let/ denote the number of the column in which it lies. Dividing /j by/ and calling the quotient q and the remainder r, we have y^(^) = 2(X)/(\) + r(\). Accordingly, if to the elements of the yth column we add those of the first, each multiplied by —qiX)-, we get an equivalent matrix in which the first element of the yth column is r(A,), which is a polynomial of degree lower than /(X). If now we interchange the first and jt\\ columns, the truth of our lemma is established in the case we are considering. A similar proof obviously applies if there is an element in the first column which is not divisible by/(X). Finally, suppose every element of the first row and column is divisible by/(X), but that there is an element, say in the ith. row and /th column, which is not divisible by/(X). Let us suppose the ele- ment in the first row and yth column is •<|r(X)/(X), and form an equivalent matrix by adding to the elements of the yth column — •</r(X) times the corresponding elements of the first column. In this matrix, /(X) still stands in- the upper left-hand corner, the first element of the yth column is zero; the first element of the z'th row has not been changed and is therefore divisible by /(X) ; and the element in the ith row and yth column is still not divisible by /. Now form another equivalent matrix by adding to the elements of the first column the corresponding elements of theyth column. The upper left-hand element is still /(X), while the first element of the * By the first element of a matrix we will understand the element in the upper left- hand comer ELEMENTARY DIVISORS AND A.-MATRICES 265 zth row is not divisible by f(X). This matrix, therefore, comes under the case already treated in which there is an element in the first column which is not divisible by/(X), and our lemma is established. Lemma 3. If we have a \-matrix whose elements are not all iden- tically zero, an equivalent matrix can he formed which has the following three properties : (a) The first element f{\) is not identically zero. (6) All the other elements of the first row and of the first column are identically zero. (e) Every element neither in the first row nor in the first colwmn is divisible by f{X). For we may first, by an interchange of rows and of columns, bring into the first place an element which is not identically zero. If this is not a factor of all the other elements, we can, by Lemma 2, find an equivalent matrix whose first element is of lower degree and is not identically zero. If this element is not a factor of all the others, we may repeat the process. Since at each step we lower the degree of the first element, there must, after a finite number of steps^ come a point where the process stops, that is, where the first element is a factor of all the others. We can then, by using transformations of type (c) (Definition 1), reduce all the elements in the first row and in the first column except this first one to zero, while the other elements remain divisible by the first one. Thus our lemma is established. Finally, we note that since /(X) in the lemma just proved is a , factor of all the other elements of the simplified matrix, it must, by Theorem 1, be the greatest common divisor of all the elements of the original matrix. The lemma just proved tells us that the \-matrix of the nth. order of rank r > (1) "11 *nl *lm can be reduced by means of elementary transformations to the form (2) /,(\) b,. "l,n-l ^ ^»-i,i ■■■ ^«-x,»-i 266 INTRODUCTION TO HIGHER ALGEBRA where /i(A.) 5^ and where //A.) is a factor of all the 5's. The last written matrix being necessarily of rank r, the matrix of the (m.— l^th order (3) ''n K »-i "n-l, 1 "n-l, n— 1 is of rank r—\. Consequently, if r>l, (3) may be reduced by means of elementary transformations to the form fix) ... c. (4) "11 ''B-2, n-2 where/gCX) ^ and where/2(\) is a factor of all the c's. By Theorem 1, f^iX), being the greatest common di v isor of all the elements of (4), is also the greatest common divisor of all the 6's,and is therefore divisible by/j(\). Now it is important to notice that the elementary transformations which carry over (3) into (4) may be regarded as elementary transforma- tions of (2) which leave the first row and column of this matrix unchanged. Thus by a succession of elementary transformations, we have reduced (1) to the form (5) where neither /j nor f^ vanishes identically, /j is a factor of f^, and /j is a factor of all the c's. If r > 2, we may treat the {n — 2)-rowed matrix of the e's, wjiich is clearly of rank r — 2, in a similar manner. Proceeding in this way, we finally reduce our matrix (1) to the form /i(\) ••• ■•• f^x) ... 0-0 ) . f,{X) . c,, . • .. ■ ■ ''1, 11-2 «n-2,l • ■ '^n-2, n-2 (6) ELEMENTARY DIVISORS AND \-MATRICES 267 where none of the/'s is identically zero, and each is a factor of the next following one. So far we have used merely elementary transformations of the forms (a) and (c), Definition 1. By means of transformations of the form (6) we can simplify (6) still further by reducing the coefficient of the highest power of X in each of the polynomials f^X) to unity. We have thus proved the theorem : Theorem 2. Every X-matrix of the nth order and of rank r can be reduced hy elementary^ transformations to the normal form ^i(X) ••• •■• JE^{X) ■■■ ■■• (7) Ur{X) ••• ■•• ••• where the coefficient of the highest power of X in each of the polynomials I1,{X) is unity, and Ui{X) is a factor of JEi_^^{X)for ^■ = 1, 2, ■■■ r — 1. By Theorem l,the greatest common divisor of the ^■-rowed determi- nants {i^r) of the original matrix is the same as the greatest common divisor of the i-rowed determinants of the normal form (7) to which it is reduced. These last mentioned i-rowed determinants are, however, all identically zero except those which are the product of i of the ^'s. Let (8) i:4x)U,^(x) - i:,(x) be any one of these, and suppose the integers ^j, k^, •■• ^,- to have been arranged in order of increasing magnitude. We obviously have ^j > 1, ^2 = 2) • • • kf ^ i. Consequently JE^ is a factor of U^,^, E^ of ^^^, etc. Thus I]lx)I!lX) ••• Elx) is seen to be a factor of (8), and, being itself one of the z-rowed de- terminants of (7), it is their greatest common divisor. That is, Theorem 3. The greatest common divisor of the i-rowed determi- nants of a X-matrix of rank r, when i ^ r, is DlX) = ElX)E4X) - EiX), where the E's are the elements of the normal form (7) to which the given matrix is equivalent. It may be noticed that this greatest common divisor is so deter- mined that the coefficient of the highest power of X in it is unity. 268 INTRODUCTION TO HIGHER ALGEBRA We come now to the fundamental theorem: Theorem 4. A necessary and sufficient condition for the equiva- lence of two \-matrices of the nth order is that they have the same rank r, and that for every value of i from 1 to r inclusive, the i-rowed deter- minants of one matrix have the same greatest common divisor as the i-rowed determinants of the other. To say that this is a necessary condition is, merely to restate Theorem 1. To prove it sufficient, suppose both matrices to be reduced to the normal form (7), where we will distinguish the normal form for the second matrix by attaching accents to the ^'s in it. If the conditions of our theorem are fulfilled, we have, by Theorem 3, _gT,(■^^J ^ ^^^^^^ and, since none of these ^'s are identically zero, it follows that II[{X) = M^) {i=l,2,...r). Thus the normal forms to which the two \-matrices can be reduced are identical, and hence the matrices are equivalent, since two X-matrices equivalent to a third are equivalent to each other. EXERCISES Reduce the matrix A. 1 \ A A-1 A.-1 by means of elementary transformations to the normal form of Theorem 2. Verify the result by finding the greatest common divisors A(A.) first directly, and secondly from the normal form. 2. By an elementary transformation of a matrix all of whose elements are integers is understood a transformation of any one of the following forms : (n) The interchange of two rows or of two columns. (b) The change of sign of all the elements of any row or column. (c) The addition to the elements of one row (or column) of the products of the corresponding elements of another row (or column) by one and the same integer. Starting from this definition, develop the theory of matrices whose elements are integers along the same lines as the theory of X-matrices was developed in this section. ELEMENTARY DIVISORS AXD X-MATRICES 269 92. Invariant Factors and Elementary Divisors. In place of the invariants J)i{X) of the last section,, it is, for most purposes, more convenient to introduce certain other invariants to which we will give the technical name invariant factors. As a basis for the defini- tion of these invariants we state the following theorem, which is merely an immediate consequence of Theorem 3, § 91: Theorem 1. The greatest common divisor of the i-rowed determi- nants (i= 2, 3, ••• r) of a X-matrix of rank r is divisible hy the greatest common divisov of the (i— l)-rowed determinants of this matrix. Definition 1. If a. is a X-matrix of rank r, and BIX) (^•=l, 2, ... r) the greatest common divisor of its i-rowed determinants so determined that the coefficient of the highest poiver of X is unity; and if Dq(X)=1; then the polynomial (1) i:ix)^^^^ (z = i,2, ...^) is called the ith invariant factor of a.. This definition shows that these ^'s are really invariants since they are completely determined by the Z>'s which we proved to be invariants in § 91. Moreover, by multiplying together the first i of the relations (1), we get the formula (2) DlX) = U,{X)II^{X) - KiX) (i=l, 2, ... r). This shows us that the ^'s completely determine the i)'s, and since these latter were seen in § 91 to form, together with the rank, a complete system of invariants, the same is true of the ^'s. That is. Theorem 2. A necessary and sufficient condition that two X-ma- trices be equivalent is that they have the same rank r, and that the inva- riant factors of cue he identical respectively with the corresponding invariant factors of the other. Since, in the case of a non-singular matrix of the wth order, i)„(X) differs from the determinant of the matrix only by a constant factor, we see that in this case the determinant of the matrix is, except for a constant factor, precisely the product of all the invari- ant factors. This is the case which is of by far the greatest impor- tance, and the term invariant factor comes from the fact that the E\ are really factors of the determinant of the matrix in this case. 270 INTRODUCTION TO HIGHER ALGEBRA A reference to Theorem 3, § 91, shows that our invariant factors are precisely the polynomials E^ which occur in the normal form of Theorem 2, § 91 ; and, since in that normal form each ^ is a factor of the next following one, we have the important result. Theorems. If IE-^{X), ••• EJ(K)areth,e successive invariant fac- tors of a X-matrix of rank r, then each of these E's is a factor of the next following one. This theorem enables us to arrange the invariant factors of a X-ma- trix in the proper order by simply arranging them in the order of increasing degree, two ^E/'s of the same degree being necessarily identical. The invariant factors (like the Z>'s of the last section) may be spoken of as rational invariants of our X-matrix since they are formed from the elements of the X-matrix by purely rational processes, namely the elementary transformations of § 91, which involve'only the rational operations of addition, subtraction, multiplication, and divison. In distinction to these the elementary divisors, first intro- duced by Weierstrass, are, in general, irrational invariants.* These we now proceed to define. Definition 2. If & is a X-matrix of rank r, and D^iX) is the greatest common divisor of the r-rowed determinants of a, then the linear ^ X—a,X — a',X — a", ■■■ of D^X) are called the linear factors o/a.f Since, by formula (2), I)^(X) is the product of all the invariant factors of a, it is clear that each invariant factor is merely the prod- uct of certain integral powers, positive or zero, of the linear factors of a. We may therefore lay down the following definition : * German writers, following Frobenius, use the term eltmentm-y divisor to cover ■both kinds of invariants. This is somewhat confusing, and necessitates the use of modifying adjectives such as simple elementary divisors for the elementary divisors as originally defined by Weierstrass, composite elementary divisors for the E'a. On the other hand Bromwich (Quadratic Forms and their Classification by Means of Invariant-factors, Cambridge, England, 1906) proposes to substitute the term invari- ant factor for the term elementary divisor. Inasmuch as this latter term is wholly appropriate, it seems clear that it should be retained in English as well as in German in the sense in which Weierstrass first used it. t It will be noticed that if a is non-singular, the linear factors of a are simply the linear factors of the determinant of a. ELEMENTARY DIVISORS AND X-MATRICES 271 Definition 3. Let a. be a X-matrix of rank r, and \ — «, \—a'. A,— «", ■•• its distinct linear factors. Then if Ul\)=(X-a)%\-a'yi{X-a"y^ - (i=l, 2, - r), are the invariant factors of a., such of the factors (X-afi, {X-a)\ (X-ayr, (X-a'yi, (X-aJL, (X-a')K, (\-«")«i', (\-«"y5', lx-a"Yr', as are not mere constants are called the elementary divisors of a, each elementary divisor being said to correspond to the linear factor of which it is a power* Since the invariant factors completely determine the elementary divisors and vice versa, it is clear that the elementary divisors are not merely invariants, but that, together with the rank, they form a complete system of invariants. That is. Theorem 4. A necessary and sufficient condition that two X-ma- trices be equivalent is that they have the same rank and that the elemen- tary divisors of one be identical respectively with the corresponding elemeritary divisors of the other. By means of Theorem 3 we infer the important result : Theokbm 5. The degrees e^ of the elementary divisors correspond- ing to any particular linear factor satisfy the inequalities ei^e^_, (1 = 2, 3,... r). By means of this theorem we can arrange the elementary divisors corresponding to any given linear factor in the proper order by simply noticing their degrees. * It wiU be seen that the definition just given is equivalent to the following one, in which the conception of invariant factors is not introduced : Definition. Let \—abe a linear factor of the \-matrixa of rank r, and let U be the exponent of the highest power of\—a which is a factor of all the i-rowed determi- nants (i^r) of a. If the integers e; (which are necessarily positive or zero) are defined by the formula ei=li-li-i (i=l. 2, ■•• r), then such of the expressions n_fj^Yi (\—aY' ■■■ ('\ — aY'' as are not constants are called the elementary divisors of a which correspond to the linear factor \—a. 272 INTRODUCTION TO HIGHER ALGEBRA EXERCISES 1. If <^ = and i/r = are two conies of which the second is non-singular, show how the number and kind of singular conies contained in the pencil <^ — Aj^ = depends on the nature of the elementary divisors of the matrix of the quadratic form <f> — Xf. 2. Extend Exercise 1 to the case of three dimensions. 3. Apply the considerations of this section to matrices whose elements are integers. (Cf. Exercise 2, §91). 93. The Practical Determination of Invariant Factors and Elemen- tary Divisors. The easiest general method for determining the invariant factors of a particular X-matrix is to reduce it by means of elementary transformations to the normal form of Theorem 2, §91, following out step by step the reduction used in the proof of that theo- rem. From this normal form the invariant factors may be read off ; and from these the elementary divisors may be computed, although only, in general, by the solution of equations of more or less high degree. There are, however, many cases of great importance in which the elementary divisors may more easily be obtained by other methods. The most obvious of these is to apply the definition of elementary divisors directlj^ to the case in hand. As an illustration, we mention a matrix of the wth order which has a — X as the element in each place of the principal diagonal, while all the other elements are zero except those which lie immediately to the right of or above the elements of the principal diagonal, these being all constants different from zero : - (1) a — X a — Xe„ • • • • ■■ a-X e„_i •• a-X (ClC2---<?„.i#0). The determinant of this matrix is (a — X)". The determinant obtained by striking out the first column and the last row is Ci<'2 "■ ^»-i- Accordingly Z)„(X) = (X-«)% 2)^i(X) = l, UJX) = {X-ay. Thus we see that (X — a)" is the only elementary divisor of this matrix, while the invariant factors are (X — a)" and n — 1 Vs. ELEMENTARY DIVISORS AND X-MATRICES 273 This direct method may some.times be employed to advantage in oonjunction with the method of reduction by elementary transfor- mations. Cf. Exercise 1 at the end of this section. A further means of recognizing the elementary divisors in some special cases is furnished by the following theorems whose proofs, which present no difficulty, we leave to the reader: Theorem 1. If all the elements of a \-matrix are zeros except those in the principal diagonal, and if each element of this diagonal which is not a constant is resolved into the product of a constant by powers of distinct linear factors of the form X, — a, X — a', .■■, then these powers of linear factors will be precisely the elementary divisors of the matrix. Theorem 2. If all the elements of a. \-matrix are zeros except those which lie in a certain number of non-overlapping principal minors, then the elementary divisors of the matrix may be found by taking the elementary divisors of all these principal minors. The proof of this theorem consists in reducing the given matrix to the form referred to in Theorem 1 by means of elementary trans- formations each of which may be regarded as an elementary trans- formation of one of the principal minors in question. It should be noticed that this theorem would not be true if the words invariant factors were substituted in it for elementary divisors; cf. Exercise 3 below. The invariant factors may, however, be com- puted from the elementary divisors when these have been found. 1. Prove that the matrix EXERCISES X-a x-a. x-a -1 0-1 0-1 1 1 x-a X-a X-a is equivalent to - 1 -1 -1 (A._«)2+^2 1 (A- - ay + /82 1 (\- ■ay^p^ 274 INTRODUCTION TO HIGHER ALGEBRA and hence that its elementary divisors are [A - (a + /30]=, [X - (« - ^0]'- 2. Generalize Exercise 1 to matrices of order 2n. 3. Find (a) the elementary divisors, and (b) the invariant factors of the X.\\ - 1)2 A.(X - 1)3 X-1 A 4. Determine the invariant factors and the elementary divisors of the matrix matrix 2X 3 1 X 4A. 3(\ + 2) A + 2 2X 6\ X 2X X-1 X-1 3(\ - 1) 1- A 2(A.-1) Is this matrix equivalent to the matrix in the exercise at the end of § 91 ? 5. Devise a convenient rational process for computing the invariant factors of matrices of the kinds considered in Theorems 1 and 2. 94. A Second Definition of the Equivalence of \-Matrices. The definition of equivalence of \-matrices whicli we have used so far rests on the elementary transformations. These transformations are of such a special character that this definition is not convenient for most purposes. We now give a new definition which we will prove to be coextensive with the old one. Definition. Two n-rowed X-matrices a and b are said to he equiv- alent if there exist two non-singular n-rowed \-matrioes c and d whose determinants are independent of X, and such that (1) bscad.* Since the matrices c and d have, by hypothesis, constant determi- nants, the inverse matrices c"^ and d~^ will also be X-matrices, and not matrices whcse coefficients are fractional rational functions of X as would in general be the case for the inverse of X-matrices. Consequently, if we write (1) in the form (2) a = c-^bd-i, we see that the relation established by our definition between the matrices a and b is a reciprocal one, as is implied in the wording of the definition. * We use here and in what follows the sign s between two X-matrices to denote isnat every element of one matrix is identically equal to the corresponding element of the other. ELEMENTARY DIVISORS AND A-MATRICES 275 In order to justify the definition just given, we begin by estab- lishing the Lemma. If a and b are n-rowed X-matrices, and the polynomial ^(\) is a factor of all the i-roued determinants of a., it is a factor of all the i rowed determinants of ab and also of ba. For, iby Theorem 5, § 25, every i-rowed determinant of ab and also of ba is a homogeneous linear combination of certain i-rowed determinants of a. Theorem 1. If a and b are equivalent according to the definition of this section, they are also equivalent according to the definition of § 91. For in this case there exist two non-singular X-matrices, c and d, whose determinants are constants, such that relation (1) holds. Consequently, by Theorem 7, § 25,* a and b have the same rank r. Let D/X) be the greatest common divisor of the i'-rowed determi- nants of a, where i ^ r. By our lemma, D^{X) is a factor of all the i-rowed determinants of ca, and therefore, applying the lemma again, it is a factor of all the i-rowed determinants of cad, that is, of b. We can infer further that I>i{\) is the greatest common divisor of the i-rowed determinants of b. For applying to relation (2) the reasoning just used, we see that the greatest common divisor of the I'-rowed determinants of b is a factor of all the z'-rowed determi- nants of a, and cannot therefore be of higher degree than -D/X). A reference to Theorem 4, § 91, now shows us that a and b are equivalent according to the definition of that section. Theoeem 2. ^ a. and b are equivalent according to the definition of § 91, they are also equivalent according to the definition of the pres- ent section. We begin by showing that if we can pass from a matrix a to a matrix aj by means of an elementary transformation, one of the fol- lowing relations always holds : (3^ aj = ca or a^ = ad where c and d are non-singular matrices whose determinants are independent of \. To prove this we consider in succession the elementary transformations of the forms which were called (a), {h\ (c), in Definition 1, § 91. * How is it that we have a right to apply this theorem to X-matrir.es ? 276 INTRODUCTION TO HIGHER ALGEBKA (a) Suppose we interchange the ^th and qth rows. This can be effected by forming the product ca where the matrix c may be obtained by interchanging the pth and g'th rows (or columns) in the unit matrix ••• Similarly the interchange of the ^th and qth columns of a may be effected by forming the product ac, where c has the same meaning as before. In each of these cases, c may be regarded as a non-singular X,-matrix with constant determinant, since its elements are constants and its determinant is — 1. (b) To multiply the pth row of a by a constant k, we may form the product ca, where c differs from the unit matrix only in having k instead of 1 as the'^th element of the principal diagonal. Similarly, we multiply the pth column of a by k, by forming the product ac, where c has the same meaning as before. If we take the constant k different from zero, c may be regarded as a non-singular \-matrix with constant determinant. (c) We can add to the pth row of a <^(^.) times the qth row by forming the product ca, where c differs from the unit matrix only in having ^(X.) instead of zero as the element in the pth row and qth column. Similarly we add to the ^th column ^(X) times the pth columa by forming the product ac where c has the same meaning as before. The matrix c, whose determinant is 1, is a non-singular X-matrix. It being thus established that one of the relations (3) holds between any two X-matrices which can be obtained from one another by an elementary transformation, it follows that two matrices a and b which are equivalent according to the definition of § 91 will satisfy a relation of the form b = Cj,Cp_i •••Ciadjclj — dj where each of the c's and d's is a non-singular X-matrix of constant determinant which corresponds to one of the elementary transforma- ELEMENTARY DIVISORS AND A-MATRICES 277 tions we use in passing from a to b. This last relation being of the form b = cad, where c and d are non-singular X-matrices with constant determi- nants, our theorem is proved. We have now completed the proof that our two definitions of the equivalence of Vmatrices are coextensive. EXERCISES 1. If a. denotes the matrix in Exercise 1, § 91, and b the normal form of Theorem 2, § 91, for this matrix, determine two X-matrices, c and d, such that relation (1) holds. Verify your result by showing that the determinants of c and d are constants. 2. Apply the considerations of this sebtion to matrices whose elements are integers. Cf. Exercise 2, § 91, and Exercise 3, § 92. 95. Multiplication and Division of X-Matrices. We close this chapter by giving a few developments of what might be called the elementary algebra of X-matrices. Definition. £i/ the degree of a \-matrix is understood the high- est degree in X of any one of its elements. For a X-matrix of the kth degree, the element in the t'th row andjth column may be written ^j.^.x* + a'yX'^-i- ^ 1- «», and at least one of the coefficients of X* {i.e. one of the %'s) musi be different from zero. If, then, we denote by a^ the matrix of which a,^' is the element which stands in the ith row and /th col- umn, we get the theorem Theokem 1. Mery Xmatrix of the kth degree may he written in the form (1) aoX*-l-aiX*-i + -4-a, (ao=?^0) where a,,, ■ ■ • a^ are matrices with constant elements ; and conversely., every expression (1) is a X-matrix of .degree h. Theorem 2. The product of two X-matrices of degrees k and I aoX* + aiX*-i+-H-a, (ao^O) boX'-l-biX'-i-|---f-b, (bo#0) is a X-matrix of degree k -1- I provided at least one of tJie matrices a^ and bfl is no n- singular. 27S INTRODUCTION TO HIGHER ALGEBRA For this product is a X-matrix of the form where Cq has the value aob,, or hQ&Q according to the order in which the two given matrices are multiplied together. By Theorem 7, § 25, neither aQbp nor b^aQ is zero if slq and b^ are not both singular. The next theorem relates to what we may call the division of X-matrices. » Theorem 3. if a and b are two \-matrices and if b. when written in the form (1), has as the coefficient of the highest power of X a non- singular matrix, then there exists one, and only one, pair of X-matrices qj and r j for which a = a b 4- r and such that either r^ s 0, or tj is a \- matrix of lower degree than b/ and also one and only one pair of X-matrices qg and x,^for which a = bqa + r^ and such that either r^ = 0, or i^ is a X-matrix of lower degree than b. The proof of this theorem is practically identical with the proof of Theorem 1, § 63. EXERCISE Definition. By a real matri^ris understood a matrix whose elements are real; by a real X-matrix, a matrix whose elements are real polynomials in X; and by a real elementary transformation, an elementary transformation in which the constant in (J) and the polynomial in (c), Definition 1, § 91, are real. Show that all the results of this chapter still hold if we interpret the words matrix, X-matrix, and elementary transformation to mean real matrix, real X-tnatrix, and real elementary transformation, respectively. CHAPTER XXI THE EQUIVALENCE AND CLASSIFICATION OF PAIRS OF BILINEAR FORMS AND OF COLLINEATIONS 96. The Equivalence of Pairs of Matrices. The applications ot the theory of elementary divisors with which we shall be concerned in this chapter and the next have reference to problems in which \-matrices occur only indirectly. A typical problem is the theory of a pair of bilinear forms. The matrices a and b of these two forms have constant elements, and we get our X-matrix only by consider- ing the matrix a — Xb of the pencil of forms determined by the two given forms. It will be noticed that this matrix is of the first degree, and in fact we shall deal, from now on, exclusively with \-matrices of the first degree. By the side of this simplification, a new difficulty is introduced, as will be clear from the following considerations. We shall subject the two sets of variables in the bilinear forms to two non-singular linear transformations whose coefficients we naturally assume to be constants, that is, independent of X. These transformations have the effect of multiplying the X-matrix, a — Xb, by certain non-singular matrices whose elements are constants (cf. § 36) and therefore, by § 94, carry it over into an equivalent X-matrix which is evidently of the fiist degree. The transformations of § 94, however, were far more general than those just referred to, so that it is not at all ob- vious whether every X-matrix of the first degree equivalent to the given one can be obtained by transformations of the sort just re- ferred to or not. These considerations show the importance of the following theorem : Theorem 1. If aj, &^, b,., bj are matrices with constant elements of which the last two are non-singular, and if the \-matrices of the first degree mi = a^ - Xbj, m^^A^- Xbg 279 280 INTRODUCTION TO HIGHER ALGEBRA are equivalent, then there exist two non-singular matrices, p and q., whose elements are independent of \, and such that (1) m2 = pmiq. Since mj and m.^ are equivalent, there exist two non-singular X-matrices, p^ and q^, whose determinants are constants and such that (2) m2 = Pomiq9. The matrix q^ has, therefore, an inverse, qp^ which is also a , X,-matrix. Let us now divide pg by m^ and q^ ^ by nij by means of Theorem 3, § 95, in such a way as to get matrices pj, p, Sj, s which satisfy the relations (3) PosmaPi + p, qo^sSimi + s, p and s being matrices whose elements are independent of \. From (2) we get _, Substituting here from (3), we have or (4) niaCPi - Si)mi = mj s - pmi- From this identity we may infer that pj = Sj and therefore <5) rngsspm^. For if Pj — Si were not identically zero, mjCPj — Sj) would be a X-matrix of at least the first degree (cf . Theorem 2, § 95), and hence the left-hand side of (4) would be a \-matrix of at least the second degree. But this is impossible, since the right-hand side of (4) is a A,-matrix of at most the first degree. If we knew that p and s were both non-singular, our theorem would follow at once from (5) ; for we could write (5) in the- form (6) m2 = pmis-i and p and s~^ would be non-singular matrices with constant elements. Moreover, we see from (5) that p and s are either both singular or both non-singular. Our theorem will thus be proved if we can show that s is non-singular. PAIRS OF BILINEAR FORMS 281 For this purpose let us substitute in the identity for q-i its value from (3), 0) I = qflSimi + QoS. Now divide q,, by ra^ by means of Theorem 3, § 95, in such a way as to get where q is a matrix with constant elements. Substituting this value in (7), we have I s qoSiHii + qjiUaS + qs. Referring to (5), we see that this may be written (9) I-qs = (qoSi + qip)mi. From this we infer that q^Sj + qjp must be identically zero, and therefore (10) I = qs. For if qgSj + qjp were not identically zero, the right-hand side of (9) would be a X-matrix of at least the first degree, while the left- hand side of (9) does not involve X. Equation (10) shows that s is non-singular, and thus our theorem is proved. It shows us, however, also that q is non-singular, and that q = s~^, so that equation (6) becomes nig = pm^q. We may, therefore, add the following CorolTjARY. The matrices p and q whose existence is stated in the above theorem may he obtained as the remainders in the division of pg and qQ in (2) by m.^ by means of theformuloe : Po - °i2Pi + P' ^0 = qi™2 + 1- From this theorem concerning Vmatrices of the first degree we .can now deduce the following theorem concerning pairs of matrices with constant elements. It is this theorem which forms the main foundation for such applications of the 'theory of elementary divisors as we shall give. We shall naturally speak of two pairs of matrices with constant elements ai, bj and a^, \ as equivalent if two non-singular matrices p and q exist for which (11) a2 = paiq, b2 = pbiq. 282 INTRODUCTION TO HIGHKR ALGEBRA Theore:\i 2. If a-^, bj and a^, bj are two pairs of matrices independent of \, and if b^ and bg are nan singular, a necessary and sufficient condition that these two pairs of matrices be equiva- lent is that the two X-matrices nij = aj — Xbj, nig = 3^ — ^^2 have the same invariant factors, — or, if we prefer, the same elementary divisors. For if the pairs of matrices are equivalent, equations (11) hold; hence, multiplying the second of these equations by X and subtracting it from the first, we have (12) m2 = pmiq, that is the X,-matrices nij and m^ are equivalent, and therefore have the same invariant factors, and the same elementary divisors. On the other hand, it follows at once from the assumption that b^ and bg are non-singular, that nij and la^ are non-singular, and hence have the same rank. Consequently if mj and va.^ have the same invariant factors, or the same elementary divisors, they are equivalent. Since they are of the first degree, there must, by Theorem 1, exist two non-singular matrices p and q, whose elements are independent of \, which satisfy the identity (12). From this identity, the two equa- tions (11) follow at once ; and the two pairs of matrices are equivalent. Thus the proof of our theorem is complete. A case of considerable importance is that in which the matrices bj and b2 both reduce to the unit matrix I. In this case nij and m.^ reduce to what are known as the characteristic matrices of aj^ and Ej respectively, according to the following definition : Definition. If a. is a matrix of the nth order with constant ele- ments and I the unit matrix of the nth order, the \-matrix A s a - XI is called the characteristic matrix of a; the determinant of A is called the characteristic function of a; and the equation of the nth degree in \ formed by setting this determinant equal to zero is called the char- acteristic equation of a. PAIRS OF BILINEAR FORMS 283 We can now deduce from Theorem 2 the following more speci'^^ result : Theorem 3. If Si^ and a^ are two matrices independent ofX,a neces- sary and sufficient condition that a non-singular matrix p exist such that * (13) a2 = pajp-i is that the characteristic matrices Aj and A^ of aj and a.^, ^ave the same invariant factors, — or, if we prefer, the same elementary divisors. For if Aj and Aj have the same invariant factors (or elementary divisors), there exist, by Theorem 2, two non-singular matrices p and q such that a2 = paiq, I = plq. The second of these equations shows us that q = p"^ ; and this value being substituted in the first, we see that p is the matrix whose existence our theorem asserts. That, on the other hand, Aj and A2 have the same invariant factors and elementary divisors if equation (13) is fulfilled, is at once obvious. 97. The Equivalence of Pairs of Bilinear Forms. Suppose we have a pair of bilinear forms in 2 n variables and also a second pair <j,^ = ia!i^Xiyj, f^ = -Lb'^Xiyj; and let us assume that -fj and yjr^ are non-singular. "We will in- quire under what conditions the two pairs of forms are equivalent, that is, under what conditions a first non-singular linear transforma- tion for the x'a and a second for the y's, :::::.: d : ; : ; : : ,x„ = c^^x[ 4- •■• + c„^x'„ [yn= <iy'i + ••■ + d^^y'„ can be found which together carry over ^j into ^^ and i/fj into i/r^. « Two matrices connected by a relation of the form (13) are sometimes called gmi7ar matrices. This conception of similarity is evidently merely a special case of the general conception of equivalence as defined in § 29, the transformations considered being of the form (13) instead of the more general form usually considered in this chapter and the last. 284 INTRODUCTION TO HIGHER ALGEBRA If we denote the conjugate of the matrix c by c' and the matrices of (^1, iItj, <f)^, 1^2 by aj, bp a^, bj respectively, we know, by Theorem 1, § 36, that the transformations c, d carry over <^j and ■\jrj^ into forms with matrices c'ajd, c'bjd respectively; so that, if these are the forms ^^ ^^^ "^t ^® have (1) aj = c'ajd, h^ = c'bjd. Consequently, by Theorem 2, § 96, the two \-matrices a^ — Xbj, 3-2 — Xbj have the same invariant factors and elementary divisors. Conversely, by the same theorem, if these two X-matrices have the same invariant factors (or elementary divisors), two constant matrices c' and d exist which satisfy both equations (1); and hence there exists a linear transformation of tlie x's and another of the «/'s which together carry over <^j into (f)^ and t/t^ into yjr^. Thus we have proved the Theorem. If <j>p yjr-^ and (j)^-, i/^g '*''* ^^^ ^<^*''*s of bilinear forms in 2 n variables of which -x/tj and ^jr^ are non-singular, a necessary and sufficient condition that these two pairs of forms be equivalent is that the matrices of the two pencils ^^ _ ^^^^ ^^ _ ^^^ have the same invariant factors, — or, if we prefer, the same elementary divisors* EXERCISE Pi-ove that the theorem of this section remaiBS- true if the' bilinear forms <^i, \pi, 4>i, \j/2 are real and the term equivalent is understood to mean equivalent with regard to real non-singular linear transformations. 98. The Equivalence of Collineations. A second important appli- cation of the theory of elementary divisors is to the theory of col- lineations. For the sake of simplicity we will consider the case of X^ = ^ji^Ij -f- a^2^2 "t" ''^IS'^S' ajg = a^^x^ -|- a^2^2 + "^33^3' although the reasoning will be s^en to be perfectly general. * For the sake of brevity, we shall, in future, speak of these invariant factors and elementary divisors as the invariant factors and elementary divisors of the pairs of forms 01, ^1 and 02, ^2 respectively. PAIRS OF BILINEAR FORMS 285 We have so far regarded a coUineation merely as a means of transforming certain geometric figures. It is possible to adopt an- other point of view, and to study the coUineation in itself with special reference to the relative -position of points before and after the transformation. Thus suppose we have a figure consisting of the points J.^ A^, ••• , finite or infinite in number, and suppose these points are carried over by the coUineation a into the points A[, A'2, ■■■ . These two sets of points together form a geometric figure. It is the properties of such figures as this that we call the properties of the coUineation. Such properties may be either projective or metrical. Thus it would be a metrical property of a coUineation if it carried over some particular pair of perpendicular lines into a pair of perpendicular lines; it would be a projective property of the coUineation if it carried over some particular triangle into itself. We shall be concerned only with the projective properties of coUineations. As an example, let us consider the fixed points of the coUineation, that is points whose initial and final position is the same. In order that (ajj, x^^, iCg) be a fixed point it is necessary and sufficient that xj = \Xi, x'2 = Xx^, x'^ = Xxg, that is, substituting in a, that a constant X exist such that, (1) fflgja;! + (a22 - X)2;2 + a^sXg = 0, «31^1 + «32 ^2 + («33 - ^>3 = ^' The matrix of this system of equations is precisely what we have called the characteristic matrix of the matrix a of the linear trans- formation. The characteristic function is a polynomial of the third degree in \ which, when equated to zero, has one, two, or three dis- tinct roots. Let Xj be one of these roots. When this is substituted in (1), these equations are satisfied by the coordinates of one or more points, — the fixed points of the coUineation a. The number and distribution of these fixed points give an important example of a projective property of a coUineation ; and it is readily seen that coUineations may have wholly different properties in this respect, one having three fixed points, another two, and still another an infinite number. Coming back now to the two sets of points J-j, A^-, ••• and A[, AL ■■■ which correspond to one another by means of the coUinea- 286 INTRODUCTION TO HIGHER ALGEBRA tion a (which may be singular or non-singular), let us subject all these points to a non-singular coUineation c, which carries over A-^, A^, ••• into B^, B^, ■■■ and A\, A'^, ■■■ into B[, B'^, ■•• respectively. The fig- ure formed by the B's will have the same projective properties as that formed by the ^'s; and consequently if we can find a coUinea- tion b which carries over B^, B^, ••■ into B[, B'^, •••, this coUineation will have the same projective properties as the coUineation a. Such a coUineation is clearly given by the formula (2) b = cac-i since c^ carries over the points B^ into the points Ai, a then carries over these into A^, and c carries over the points A'^ into the points jBJ. Since two coUineations a and b related by formula (2) are indis- tinguishable so far as their projective properties go (though they may have very different metrical properties), we will call them equivalent according to the following Definition. Two coUineations a and b shall be called equivalent if a non-singular coUineation c exists such that relation (2) is fulfilled. A reference to Theorem 3, § 96, now gives us the fundamental theorem : Theorem. A necessary and sufficient condition that two coUinea- tions be equivalent is that their characteristic matrices have the same in- variant factors, — or, if we prefer, the same elementary divisors. EXERCISES 1. If Pi, i-'2i •■• Pi are fixed points of a non-singular coUineation in space of n — 1 dimensions which correspond to k distinct roots of the characteristic equa- tion, prove that these points are linearly independent. 2. Discuss the distribution of the fixed points of a coUineation (a) in two dimensions, (6) in three dimensions, lOr all possible cases of non-singular coUineations, 3. Discuss the distribution of (a) the fixed lines of a coUineation in two dimensions, (J) the fixed planes of a coUineation in three dimensions, for all possible cases of non-singular coUineations ; paying special attention to their relation to the fixed points. 4. Two real coUineations, a and b, may be said to be equivalent if there exists a real non-singular coUineation c such that b = cac-i. With this understanding of the term equivalence, show that the theorem of the present section holds for real coUineations. PAIRS OF BILINEAR FORMS 287 99. Classification of Pairs of Bilinear Forms. We consider again the pair of bilinear forms of which we assume the second to be non-singular, and form the \-matrix. (1) a-Xb. Using a slightly different notation from that employed in § 92, we will denote the elementary divisors of (1) by (\ - Xj^i, (X - X^)% ...... (X - X,)«*, (ej + e^ + • • . + e* = n), so that the linear factors X — X^ need not all be distinct from one another. The most important thing concerning these elementary divisors is, for many purposes, their degrees, ej, e^, ■■■ e/,. When we wish to indicate these degrees without writing out the elementary divisors in full, we will use the symbol [gj e^ ■■■ e^'], called the char- acteristic of the X-matrix (1), or of the pair of forms (j), ■^. It will be seen that this characteristic is a sort of arithmetical invariant of the pair of bilinear forms, since two pairs of bilinear forms which are equivalent necessarily have the same characteristic. The converse of this, however, is not true, since for the equivalence of two pairs of bilinear forms the identity of the elementary divisors themselves, not merely the equality of their degrees, is necessary. All pairs of bilinear forms which have the same characteristic are said to form a category. Thus, for example, in the case of pairs of bilinear forms in six variables we should distinguish between three categories corresponding to the three characteristics, [1 1 1], [2 1], [3], which are obviously the only possible ones in this case. In fact, we must inquire whether these three categories really all exist. This question we answer in the affirmative by writing down the following pairs of bilinear forms in six variables which represent these three categories : , , Xj - X Xj - X Xg-X 288 INTRODUCTION TO HIGHER ALGEBRA 1 ^1^1+ ^2^2 + ^32/8' Aj "^ A. 1 A.1 "^ \ III.. [3] 1 i«;i«/l + ^^2^2 + A.1 — A* Aj — A 1 The pairs of bilinear forms we have just written down do more than merely establish the existence of our three categories. They establish the fact that not only the degrees of the elementary divisors are arbitrary (subject merely to the condition that their sum be three), but that, subject to this restriction, the elementaiy divisors themselves may be arbitrarily chosen. They are, moreover, normal forms to one or the other of which every pair of bilinear forms in six variables, of which the first is non -singular, may be reduced by non-singular linear transformations. The general theorem here is this: Thboeem. If Xj, Xgi • ■ • ^/i; <*'"« o-if^y constants, equal or unequal, and gj, fig, • • • e^, are any positive integers whose sum is n, there exist pairs of bilinear forms in 2 n variables, the second form in each pair being non- singular, which have the elementary divisors (2) (x-Xi)% (x-\)% (x-x^y*. The proof of this theorem consists in considering the pair of bilinear forms (3) (n n \ .V" = a^i^i + x^y^ -f ••• -I- x„y^. PAIRS OF BILINEAR FORMS 28^ of which the second is non-singular. These forms have a X-matrix which may be indicated, for brevity, as 14) M, M, M, where the letters Mj, • • ■ M^ represent not single terms but blocks of terms ; Mj standing for the matrix of order e^ M, X,-\ 1 ••■ \(-\ 1 ... A,/ — \ while all the terms of the matrix (4) are zero which do not stand in one of the blocks of terms M,-. The elementary divisors of (4) are, as we see by a reference to § 93 (Formula (1) and Theorem 2), pre- cisely the expressions (2). Thus our theorem is proved. A reference to § 97 shows that formula (3) is a normal form to which every pair of bilinear forms in 2 m variables with the ele- mentary divisors (2) can be reduced.* * Many other normal forms might be chosen in place of (3). Thus, for instance, we might have used in place of (3) the form (3') 1 1 Cj+l «i+l n-ejt+1 n— eft+l <f' = SCiXa/e-i+l + 2C2!C<y2j +e -i+1 + ^CiXiVle^+le^+e^-i+l 1 'i+l , «i+«z+l + + 2Cir£(2/2n-e4-i+l, where the constants ci, ••• c/e, di, ••• dh may be chosen at pleasure provided, merely that none of them are zero. For instance, they may all be assigned the value 1. V 290 INTRODUCTION TO HIGHER ALGEBRA Let us now return to the classification of pairs of bilinear forms. For a given number, 2 n, of variables we have obviously only a finite number of categories. We may subdivide these categories into classes by noticing which, if any, of the elementary divisors corre- spond to the same linear factor. This we can indicate in the char- acteristic by connecting by parentheses those integers which are the degrees of elementary divisors corresponding to one and the same linear factor. Thus, in the case w = 8, the characteristic [(21)(111)2] Would indicate that the X-matrix has just three distinct linear fac- tors ; that to one of these there correspond two elementary divisors of degrees two and one respectively, to another three elementary divisors of the first degree, and to the last a single elementary divisor of degree two. Two pairs of bilinear forms which are equivalent belong neces- sarily to the same class, but two pairs of bilinear forms which be- long to the same class are not necessarily equivalent. To illustrate what has just been said, let us again consider the case w = 3. Here we have now, instead of three categories, six classes, which are exhibited in the following table : a h c I. [111] [(1 1) 1] [(111)] II. [21] [(2 1)]' III. [3] The X-njatrix of this pair of forms may be written in the form (4), where, how- ever, Mi now stands for the matrix of order et : Mi = di Ci(\i — di c(Xj - X) ^) CiCKi - X) It will be noticed that the matrices Mj, and therefore also the bilinear forms (3'), are symmetrical, a fact which will make this normal form important when we come to the subject of quadratic forms in the next chapter. Constants similar to the constants Ci and di which we have introduced in (3( might also have been introduced in (3). PAIRS OF BILINEAR FORMS 291 are The three classes la, lb, Ic form together the category I, and ..__ all represented by the normal form given for that category above, the only difference being that in class la the three quantities \, xj Xg are all distinct, in class lb two, and only two, of them are equal, while in class le they are all equal. Similarly category II is now divided into two classes, Ila and IIJ, for both of which the normal form of category II holds good, \ and X^ being, however, different in that normal form for class Ila and equal for class 116. Finally category III consists of only a single class. For some purposes it is desirable to carry this subdivision still farther. The second of our two bilinear forms, yfr, has been assumed throughout to be non-singular. The first, (f), may be singular or non-singular ; and it is readily seen that a necessary and sufficient condition that ^ be singular is that one, at least, of the constants X, which enter into the linear factors of the X-matrix be zero. Thus it will be seen that in a single class we shall have pairs of bilinear forms both of which are non-singular and others one of which is singular, and we may wish to separate into different sub-classes the pairs of forms which belong to one or the other of these two cases. Let us go a step farther in this same direction, and inquire how the rank of ^ is connected with the values of the constants X^. We notice that the matrix of <^ is equal to the matrix of the pencil <j) — \-\jr when X = 0. Accordingly, if (j) is of rank r, every (r -(- l)-rowed determinant of the matrix of ^ — \yjr will be divisible by X, while at least one r-rowed determinant of this matrix is not divisible by X. It is then necessary, as we see by a reference to the definition of elementary divisors (cf. the footnote to Definition 3, § 92), that just w — r of the constants X,- which enter into the elementary divisors should be zero. Since the converse of these statements is also true, we may say that a necessary and sufficient condition that the form <f> be of rank r is that just n— r of the elementary divisors be of the form X'i. Let us, in the characteristic [e-^^ e^ ■■■ ej, place a small zero above each of the integers e^ which is the degree of such an elemen- tary divisor ; and regard two pairs of bilinear forms as belonging to a single class when, and only when, their characteristics coincide in the distribution of these zeros as well as in other respects. Here again two equivalent pairs of forms will always belong to the same class, but the converse will not be true. 292 INTRODUCTION TO HIGHER Al.GEBRA As an illustration, let us again take the case n = S. We have now fourteen classes instead of six. [1 1 1], [(1 1)1^], [(1 1 1)], [2 1], [(2 1)], [3], (r = 3), [111], [(11)1], [21], [21], [3J, (r=2), [(11)1], [(21)], (r=l), [(ill)], (r=0). We have indicated, in each case, the rank r of the form (f). Thus in the first six cases <j) is non-singular ; in the next five it is of rank 2, etc. EXERCISES 1. Prove that there exist pairs of real bilinear forms in 2 n variables of which the second is non-singular, and which have the elementary divisors (\-Xi)S (X-Aa)''^, (X-Xi)«t (ei + «2+ - +ei = n), provided that such of these elementary divisors as are not real admit of arrange- ment in conjugate imaginary pairs. (Cf. Exercises 1, 2, § 93.) 2. Classify pairs of real bilinear forms in six variables (the second form in each pair being non-singular), distinguishing between real and imaginary elementary divisors. 100. Classification of Collineations. The classification of pairs of bilinear forms which we gave in the last section may obviously be regarded, from a more general point of view, as a classification of pairs of matrices, the second matrix of each pair being assumed to be non-singular. From this point of view it admits of applica- tion to the classification of collineations, since, as we saw in § 98, to every coUineation corresponds a pair of matrices of which one is non-singular, namely the unit matrix I and the matrix of the linear transformation. Moreover, the normal form (3) of § 99 is precisely adapted to the treatment of the more special kind of equivalence which we have to consider l^re, since the matrix of the form i/r is precisely the unit matrix. ' We may therefore state at once the fundamental theorem : Theorem 1. if \j, Xj, ••■ X^ are any constants, equal or unequal, and fij, e^, ■■■ e^ ani/ positive integers whose sum is n, there exists a col- lineation in space of n — 1 dimensions whose characteristic matrix ha» the elementary divisors (\ - Xi)% (\ - \,)S (\ - x^y*. PAIRS OF BILINEAR FORMS 293 To this we may add Theoebm 2. Every collineation of the kind mentioned in Theorem 1 is equivalent to the collineation whose matrix is. Ml M, M. where M^ stands for the matrix of order e^, Xj 1 ... M.= X, 1 \ We thus get a classification of coUineations into categories and a subdivision of these categories into classes precisely as in § 99. For instance, in the case w = 3 (coUineations in the plane), we have three categories whose characteristics and representative normal forms we give: I. [Ill] II. [21] f^l'= ^1 ^1 4- \^x^ [xl = XgaJg, x[ = Xj x^ + x^ 4 = \x^ u= \^s, h'= Xi x^ + x^ x',= X^x^ ■^H U= \^x^. III. [3] These categories we should then subdivide either into six classes as on page 290 or into fourteen classes as on page 292. This latter classification is the desirable one in this case. We proceed to give a list of these fourteen classes with a characteristic property of each. 294 INTRODUCTION TO HIGHER ALGEBRA That the normal forms of the coUineations have these properties will be at once evident, and from this it follows that all the coUinea- tions of the class have the property in question, since the properties mentioned are obviously all projective. That the properties men- tioned are really characteristic properties, that is, serve to distin- guish one class from another, can only be seen a posteriori, by noticing that no one of the properties mentioned is shared by two classes. [1 1 1] Three distinct non-collinear fixed points.* [(11)1] Every point of a certain line and one point not on this line are fixed. [(1 1 1)] The identical coUineation. [2 1] Two distinct fixed points. [(2 1)] Every point of a certain line is fixed. [3] One fixed point. In all these cases the coUineation is non-singular. The remain- ing coUineations are singular. In the next three, one point P of the plane is not transformed at all, while all other points go over on to a line p which does not pass through P, and every one of whose points corresponds to an infinite number of points. [1 1 1] There are two fixed points on p. [(1 1) 1] Every point on p is fixed. [2 1] One fixed point on p. In the next two cases one point P is not transformed at all, while all other points go over on to a line p which passes through P, and every one of whose points corresponds to an infinite number of points. [2 1] One fixed point. [3] No fixed point. The remaining coUineations are so simple that they are not merely characterized, but completely described, by the property we mention. [(1 1) 1] The points on a certain line are not transformed. All other points go over into a single point which does not lie on this line. * It should be understood here and in what follows that the fixed points which are mentioned are the only fixed points of the coUineation in question. PAIRS OF BILINEAR FORMS 295 [(2 1)] The points on a certain line are not transformed. All other points go over into a single point on this line. . _ ' [(1 1 1)] No point iu the plane is transformed. This last case is of course not a transformation at all. SXERCISES 1. Classify, in a similar manner, the projective transformations in one dimension. 2. Classify the coUineations in space of three dimensions. 3. Classify the real projective transformations in space of one, two, and three dimensions. (Cf. Exercises 1, 2, § 99.) CHAPTER XXII THE EQUIVALENCE AND CLASSIFICATION OF PAIRS OF QUADRATIC FORMS 101. Two Theorems in the Theory of Matrices. In order to jus- tify the applications we wish to make of the'theorjj of elementary divisors to the subject of quadratic forms, it will be necessary for us to turn back for a moment to the general theory of matrices. Definition. If ^{x) is a polynomial : 4>{x) = %x"' + a-^x'"-^+ ■■■ +a^.-^x + a„, then agX" + ajX""! + ••• + a^.jX + a„I is called a polynomial in the matrix x and is denoted by <f>{x).* We come now to one of the most fundamental theorems in the whole theory of matrices : Theorem: 1. If a. is a matrix, and ^(X) its characteristic func- tion, then ^g^^ ^ 0. This equation is called the Hamilton-Cayley equation. Let c be the characteristic matrix of a : c = a — \I. This being a X-matrix of the first degree, its adjoint C will be a \-matrix of degree not higher than «— 1, if n is the order of the matrix a : (1) C = C„_i\»-i + C„_2\'-2 + ... + Co. We may also write (2) <^(\) = Ar„\" + VA"-^+ - +V Now referring to formula (5), § 25, we see that aC - \C s ^(\)I. * It should be noticed that, according to this definition, the coefiScients of a poly- nomial in X are scalars. Contrast this with a X-matrix, in which the coefiScients are matrices and the variable a scalar. Both of these conceptions would be included in expressions of the form : aox'-bo + aix"-ibi + ••• + «u_ixb„_i + a„. 29B PAIRS OF QUADRATIC FORMS 297 Substituting here from (1) and (2), we have, on equating corre- sponding powers of \, - , aCj - Co = Ajl, aC2 — Cj = k^I, ftC„_i — C„_2 — «n-ll» — C„_i = k„I. If we multiply these equations in succession by I, a, a?, ••• a", and add, the first members cancel out, and we get JcqI + k-fi + k^a? + ••• + ^„a" = 0. This is precisely the equation <^(a)=0 which we wished to establish. As a means of deducing our second theorem, we next establish a lemma which relates merely to scalar quantities. Lemma. If y]r{x) is a polynomial of the nth degree (n>0) whose constant term is not zero, there exists a polynomial x(^) of degree less than n such that , , nno is divisible by yj/{x). Let X— a, x—i, x — c, •■• be the distinct linear factors of ■y}r{x), so that we may write f{x)=k(x-ay{x-hy{x-e)y — (a + /3 + 7+ ••■ =w). None of the constants a, b, c, ••• are zero, since, by hypothesis, the constant term of ■^ is not zero. Let us, further, denote by 'f'i(x) the polynomial obtained from sjr by omitting the factor (a; — a)% by yfr^^x) the polynomial obtained from ■f by omitting the factor (x—bf, etc., and finally let us form, with undetermined coefficients, the polynomials A{x) = Ao + Aj(x-a) + A^x-ay+ - +A,_i(x-ay-\ B{x) = Bo + ^i(^ - *) + ^2(* - *)" + • ■• + -^"-1^^ ~ ^^'^' C{x) = (7o + c^x -c)+C4^x-cf+ - + Q,.lx - cy-\ 298 INTRODUCTION TO HIGHER ALGEBRA From these polynomials we now form the polynomial X(a:) = A{x)ylr^{x) + B{x)f^x) + 0{x)flx) + ... whose degree can obviously not exceed w — 1. We wish to show that the coefficients -4.,-, jB,-, ••• can be so determined that this poly- nomial 'X,{x) satisfies the conditions of our lemma. Since i/r^, 1^3, ■ • • are all divisible by {x — a)*, a necessary and sufficient condition that (x(^)T — * be divisible by this factor is that the polynomial ^a>)^^A(x)f(ylr,{x)f-x be divisible by (a; — a)". We have <(j}(a) = Alfi^a-bf\a-efy ••• -a. In oi-der that <f>{x) be divisible by 2; — a it is therefore necessary and sufficient that Neither numerator nor denominator here being zero, we thus obtain two distinct values for Aq,' both different from zero. If we give to Aq one of these values, <j>{x) is divisible by a; — a. A neces- .sary and sufficient condition that it be also divisible hy(x~a^ is that <f)'(a) = 0, accents here, and in what follows, denoting differentia- tion. We shall see in a moment that this condition can be imposed in one, and only one, way by a suitable choice of A^ The condi- tion that (l){x) be divisible by {x — of is then simply ^"{a) = 0. We wish to show that this process can be continued until we have finally imposed the condition that 4>(x) be divisible by {x — ay. For this purpose we use tlie method of mathematical induction, and assume that Aq, ■•• ^j_j have been so determined that <f>(a) = <f>'{a')= ••• = <^f'~i^(a)=0. It remains then merely to show that A, can be so determined that <f>^'\ci) = 0. For this purpose we notice that (4) cl>^'Kx) = 2Al^\x)A(xJylr,(x)f + R^x) where Ii,{x) is an integral rational function with numerical coeffi- cients of 1/^1, i/rj, ... i/rW J., J.', ... ^['-11. Since A{a) = A,, A'{a) = A„ A"{a) = 21 A^, - A^'- ')(«) = (s - ly.A^^, it follows that B{a) is a known constant, that is, that it does not depend on any of the still undetermined constants A„A,^^, ... A„^^, PAIRS OF QUADRATIC FORMS 299 nor on the ^'s, 0% etc. Consequently we see from (4) that a neces- sary and sufficient condition that </)W(a) = is that A, have the value ^5) A- - RM Determining the coefficients Aj, ^^^ - A-i in succession by means of this formula, we finally determine the polynomial A{x) in such a way that <^(a;) is divisible by {x-a)-. For this determination, (xi^)f - « will, as we saw above, be divisible by {x - a)". In* the same way we can now determine the coefficients of B(x) so that (j^x))^-zis divisible by {x-hf; then we determine the coeffi- cients of C{x) so that {x{^)f - a; is divisible by {x - o)y; etc. When all the polynomials A, B, 0, ■■• are thus determined, '(X(a;))2 -a; is divisible by i/r(a;), and our lemma is proved. Theorem 2. If a. is a non-singular matrix of order n, there exist ma- trices b of order n {necessarily non-singular) with the following properties : b2 = a, b is a polynomial in a of degree less than n. Since a is non-singular, its characteristic function 4>(X) is a poly- nomial of the wth degree whose constant term is not zero. Hence, by the preceding lemma, a polynomial ;;^;(\) of degree less than n can be determined such that ^^(^^-^^ _ ^ ^ <^(X)/(\) where f(X) is also a polynomial. From this identity it follows that (X(a))2-a = </.(a)/(a). Since, by Theorem 1, ^(a) = 0, the last equation may be written (X(a))^ = a, so that b = x(a) is a matrix satisfying the conditions of our theorem, which is thus proved. 102. Symmetric Matrices. The application of the theory of ele- mentary divisors to the subject of quadratic forms rests on the fol- lowing proposition: Theorem 1. If sl^ and a.^ are symmetric matrices and if there exist two non- singular matrices p and q such that (1) a2 = paiq, 300 INTRODUCTION TO HIGHER ALGEBRA then there alto exists a non-singular matrix P such that (2) a, = P'aiP where P' is the conjugate of P.* Let us denote by p' and q' the conjugates of p and q respectively. Taking the conjugates of both sides of (1), and remembering that aj and a^, being symmetric, are their own conjugades, we get, by Theorem 6, § 22, (3) Eg = q'aiP'. • By equating the values of a2 in (1) and (3), we readily deduce the further relation (4) • (q'rpai = aip'q-^ For brevity we will let (5) U = (q'r'p, U'=p'q-', and note that U' is the conjugate of U ; cf. Exercise 6, § 25. Equa- tion (4) may then be written (6) Uai = aiU'. From this equation we infer at once the following further ones : D^ai = UajU' = ajU'^, U»ai = UaiU'2 = aiU'3, (7) V% = UaiU'*"^ = ajU'*. Let us now multiply the equations (6) and (7) and also the equa- tion aj = aj by any set of scalar constants and add them together. We see in this way that if % (U) is any polynomial in U, (8) x(U)a, = a.x(U'). * A proof of this theorem much simpler than that given in the text is the following : From (1) we infer at once that ai and a^ have the same rank. Hence the quad- ratic forms of which ai and a2 are the matrices are equivalent to each other by Theorem 4, § 46. If we denote by P the matrix of the linear transformation which carries over the quadratic form ai into the form 32, vye see, from Theorem 1, § 43, that equation (2) holds. This proof would not enable us to infer thatP can be expressed in terms of p and q alone, and this is essential for our purposes. PAIRS OF QUADRATIC FORMS 301 We will choose the polynomial V = X(U) so that V is non-singular and as is seen to be possible by Theorem 2, § 101. Denoting by V' the conjugate of V, we evidently have V' = %(U'), so that we may write (8) in the form Vai = ajV, or ai = V-iaiV'. We now substitute this value in (1) and get (9) a2 = pV-iaiV'q. From the first equation (5) we infer the formula pV-i=q'V. Consequently pV~i is the conjugate of V'q, so that if we let P = V'q, equation (9) may be written a^ = P'aiP, and our theorem is proved. The proof just given enables us to add the Corollary. As the matrix P of the foregoing theorem may he taken the matrix V'q where V' is the conjugate of any one of the square roots, determined hy Theorem 2, § 101, of (q')"^ p- \ In particular it will be seen that P depends on p and q but not on Ej or a^. Hence if aj, ag, bj, \ are symmetric matrices, and there exist two non-singular matrices p and q such that aj = pajq, \ = pbjq, then there exists a non-singular matrix P such that a2 = P'ajP, bo = P'bjP. 302 INTRODUCTION TO HIGHER ALGEBRA From this and Theorem 2, § 96, we infer Theorem 2. If &i, ag, bj, bg, are symmetric matrices of which bj, bj are non- singular, a necessary and sufficient condition that a non-singu- lar matrix P exist such that (10) 32 = P'aiP, \ = P'bjP, where P' is the conjugate of P, is that the matrices aj — Xbj, 02 — Xbj have the same invariant factors, — or, if we prefer, the same elementary divisors. If, in particular, bj = bg = I, where I is the unit matrix, we have, from the second equation (10), the formula I = P'P. Such a matrix P we call an orthogonal matrix according to the defini- tion, which will readily be seen to be equivalent to the one given in. the first footnote on page 154 : Definition. By an orthogonal matrix we understand a non-singu^ lar matrix whose inverse is equal to its conjugate. In the special case just referred to, Theorem 2 may be stated in the following form : Theorem 3. If aj and sl^ are two symmetric matrices, a necessary and sufficient condition that an orthogonal matrix P exist such that a2=P'aiP is that the characteristic matrices of aj and s^ have the same invariant factors, — or, if we prefer, the same elementary divisors. If this theorem is compared with Theorem 3, § 96, it will be seen that it differs from it only in two respects, first that a^ and a^ are assumed to be symmetric, and secondly that P is required to be orthogonal. 103. The Equivalence of Pairs of Quadratic Forms. Let us con- sider the two pairs of quadratic forms n n ^1 = 2 aljXiXj, fi = ^ K- <«iXj, n n and </>2 = 2) a'}jXiXj, V^a = ^ ^'^i^,', PAIRS OF QUADRATIC FORMS 803 of which the two forms ■^-^ and 1^2 ^'re assumed to be non-singular. We will inquire under what conditions these two pairs of forms are equivalent ; that is, under what conditions a linear transformation -nlA + •■• + C„X exists which carries over ^^ into ^^ and, at the same time, i/tj into ■^j- If we denote the conjugate of the matrix c by c', and the ma- trices of the forms <^j, t/tj,, ^gi ""/^a ^7 %' ^r ^2' \ respectively, we know, by Theorem 1, § 43, that the transformation c carries over ^j and i/tj into forms with the matrices c'ajC, c'biC respectively; so that, if these are the forms ^^ and -^g' ^^ have (1) a^ = c'eiC, \ = c'bjC. Consequently, by Theorem 2, § 102, the two \-matrices aj — Xbj, aj — Xbg have the same invariant factors and elementary divisors. Conversely, by the same theorem, if these two X-matrices have the same invariant factors (or elementary divisors), a matrix c, inde- pendent of X, exists which satisfies both equations (1) ; and hence the two pairs of quadratic forms are equivalent. Thus we have proved Theorem 1. If 0j, i/^j and (j)^, tjr^ are two pairs of quadratic forms in n variables, in which -yjr^ and -yfr^ are non-singular, a necessary and sufficient condition that these two pairs of forms he equivalent is that the matrices of the two pencils have the same invariant factors, — or, if we prefer, the same elemen- tary divisors* A special case of this theorem which is of considerable impor- tance is that in which both of the forms i/tj and i/r^ reduce to xl + xl+ - +xl. • For brevity, we shall speak of these Invariant factors and elementary divisors as the invariant factors and elementary divisors of the pairs of forms ^1, ifri and ^2, i^j respectively. 304 INTRODUCTION TO HIGHER ALGEBRA In this case we have to deal with orthogonal transformations (cf. the Definition in Exercise 1, § 52), and our theorem may be stated in the form * Theorem 2. If s.^ and a^ are the matrices of two quadratic forms, a necessary and sufficient condition that there exist an orthogonal trans- formation which carries over one of these forms into the other is that the characteristic matrices of a.^ and a^ have the same invariant factors, — or, if we prefer, the same elementary divisors. To illustrate the meaning of the theorems of this section, let us consider again briefly the problem of the simultaneous reduction of two quadratic forms to sums of squares. In Chapter XIII we be- came acquainted with two cases in which this reduction is possible ; cf. Theorem 2, § 58. and Theorem 2, § 59. We are in a position now to state a necessary and sufScient condition for the possibility of this reduction, provided that one of the two forms is non-singular. For this purpose, consider the two quadratic forms ^ = k^xl-irh^l+ ■■• +k„xl, -f = c^xl + c^l + — + c^xl. where we assume, in order that the second form may be non-singular, that none of the cs vanish. The matrix of the pencil ^ — X'^ is ^1 ~ '^i^ K — cJk and the elementary divisors of this matrix are \-'h, \-% \- Co K ^1 ^2 •-» all of the first degree. Consequently, any pair of quadratic forms equivalent to the pair just considered must have a \-matrix whose elementary divisors are all of the first degree. Conversely, if we have a pair of quadratic forms, of which the first is non-singular, whose \-matrix has elementary divisors all of * This theorem is, of course, essentially equivalent to Theorem S, § 102, of which it may he regarded as an immediate consequence. PAIRS OF QUADRATIC FORMS 305 the first degree, we can obviously choose the constants h and o in such a way that the \-matrix of the forms <f> and yjr just con- sidered has these same elementary divisors, and therefore the given forms are equivalent to these special forms ^ and yfr. Thus we have proved the theorem : Theorem S. If (f> and yjr are quadratic forms and yfr is non-singu- lar, a necessary and sufficient condition that it be possible to reduce (}> and ■yfr simultaneously by a non-singular linear transformation to forms into which only the square terms enter is that all the elementary divisors of the pair of form's be of the first degree. This theorem obviously includes as a special case Theorem 2 of § 68, since the elementary divisors are necessarily of the first degree when the X-equation has no multiple roots. Comparing the theorem just proved with Theorem 2, § 59, we see that under the conditions of that theorem the elementary divisors must be of the first degree. Hence Theorem 4. If ■^ is a non-singular, definite, quadratic form, and <f) is a real quadratic form, all the elementary divisors of this pair of forms are necessarily of the fvrst degree. 104. Classification of Pairs of Quadratic Forms. We consider the pair of quadratic forms n n (1) (j, = ^a^-XiXj, -f = '^hjXiXp and assume, as before, that -\/r is non -singular. We denote the ele- mentary divisors of these forms, as in § 99, by (\ - Xi>, (X - X2)% - (X - x^y* («! + ^2 +■•• + «* = '»)• The symbol [ej e^ •••ej we call the characteristic of the pair of quadratic forms; and all pairs of quadratic forms which have the same characteristic we speak of as forming a category.* We have here, precisely as in the case of bilinear forms, the theorem: Theorem. IfXy, X2, ••• X* are any constants, equal or unequal, and e ,e , ■■■ e^are any positive integers whose sum is n, there exist pairs * Thus, for instance, all pairs of forms of which the second is non-singular and which admit of simultaneous reduction to sums of squares, form a category whose character istic is [1 1 - 1]. Cf. Theorem 3, § 103. X 306 INTRODUCTION TO HIGHER ALGEBRA of quadratic forms in n variables, the second form in each pair being non-singular, which have the elementary divisors (2) (A,-Xj)% {X-X^)\ (X-\*y'. The proof of this theorem consists in considering the following pair of quadratic forms, analogous to the normal form (3') of § 99: ^1 t ' ^/> -1-1 p.■^-^ ' (3) n 1 ' '^1+1 «!+! + - +/ n-l B-1 \ ( 2 \^C i^iX^n-e -iJr\ + ^ */:^i^2«-fi-« j' 1 Ci+l e,+Cj+l + ■" + — ''A^i-'-an-fi-i+l' where Cj, ••• e^, tij, ■•■ d^ are constants which may be chosen at pleas- ure, provided none of them are zero. The \-matrix of this pair of forms is the same as the \-matrix of the pair pf bilinear forms (3' ) of § 99, and therefore has the desired elementary divisors. A reference to Theorem 1, § 103, shows that formula (3) yields a normal form to which every pair of quadratic forms, of which the second is non-singular and whose elementary divisors are given by (2), can be reduced. The categories, of which we have so far spoken, may be divided into classes by the same methods we used in § 99 in the" case of bilinear forms. This may be done, as before, either by simply noting which of the X/s are equal to each other, or by further distinguishing between the cases where some of the X/s are zero. We are now in a position to see exactly in what way our elemen- tary divisors give us a more powerful instrument than we had-in the invariants @, of § 57. These invariants @„ being the coefficients of the X-equation of our pair of forms, determine the constants Xj, which are the roots of this equation, as well as the multiplicities of these roots. They do not determine the degrees e^ of the elementary di- visors, and the use of the @,'s alone does not, in all cases, enable us to determine whether two pairs of forms are equivalent or not. Thus, for instance, we may have two pairs of forms with exactly the PAIRS OF QUADRATIC FORMS iJO? same invariants ©, but with characteristics [(11)11 ••■1] and [211 ... 1] respectively.* It will be seen, therefore, that the ©j's form in only a very technical sense a complete sysjtem of invariants. EXERCISES 1. Form a numerical example in the case n = 3 to illustrate the statement made in the next to the last sentence of this section. 2. Prove that if two equivalent pairs of quadratic forms have two elementary- divisors of the first degree which correspond to the same linear factor, there exist an infinite nutnber of linear transformations which carry over one pair of forms into the other. 3. Prove the general theorem, of which Exercise 2 is a special case, namely, that if two equivalent pairs of quadratic forms have a characteristic in which one or more parentheses appear, there exist an infinite number of linear transforma- tions which carry over one pair of forms into the other. 4. Prove that if two equivalent pairs of quadratic forms have a characteristic in which no parentheses appear, only a finite number of linear transformations exist which carry over one pair of forms into the other, f How are these transformations related to each other? 105. Pairs of Quadratic Equations, and Pencils of Forms or Equa- tions. J In dealing with quadratic forms, the questions of equiva- lence and classification do not always present themselves to us in precisely the form in which we have considered them in the last two sections. We frequently have to deal not with the quadratic forms themselves but with the eqvMions obtained by setting the forms equal to zero. Two such pairs of equations we shall regard as equivalent, not merely if the forms in them are equivalent, but also if one pair of forms can be obtained from the other by multiplication by constants different from zero. Let us consider two quadratic forms ^, -v/r, of which we assume, as before, that the second is non-singular, and inquire what the effect on the elementary divisors (1) {\-\)% (X - \)\ ...... {X- X.yic * We may, in the case n = 3, put the same thing geometrically (of. the next sec- tion) by saying that it is impossible to distinguish between the case of two oonics having double contact and that of two conies having simple contact at a single point by the use of the invariants &i alone, whereas these two cases are at once distinguished by the use of elementary divisors. t The exercise in § 58 is practically a special case of this. X Questions similar to those treated in this section might have been taken up in the last chapter for the case of bilinear forms. 308 INTRODUCTION TO HIGHER ALGEBRA of these forms will be if the forms are multiplied respectively by the constants p, q which are both assumed to be different from zero. Let us write Then (2) 4>^-\^^=p{4>-\'^) where \'—'Lx. P Let \ — a be any one of the linear factors of the matrix of ^ — \i^, so that a is any one of the constants Xj, X^, ■■■ Xj.; and let us denote, as in the footnote to Definition 3, § 92, by ?j the exponent of the highest power of X — a which is a factor of all the «-rowed deter- minants of this matrix. Then it is clear, from (2), that Z^ is the exponent of the highest power of X' — a which is a factor of all the t-rowed determinants of the matrix of (^j — Xi/r^. In other words, x-^y- is the highest power of the linear factor X — pa/q which is a factor of all the i-rowed determinants of the matrix of ^j — Xt/tj. Turning now to the definition of elementary divisors as given in the footnote to Definition 3, § 92, we see that the elementary divisors of the matrix of </>j — Xi/r^ differ from those of the matrix of — X->|r only in having the constants X^ replaced by the constants pX^/q. We thus have the result : Theorem 1. If the pair of quadratic forms <f>, yjr, of which the second is assumed to be non-singular, has the elementary divisors (X - x^)^, (X - \)\ (X - x^y* . and if p., q are constants different from zero, then the pair of quadratic forms pcj), q-\jr has the elementary divisors (x-x{)% (x-x^)% (x-xiy* where Xj = -X,- In particular, it will be seen that these two pairs of forms have the same characteristic, even when the conception of the character- istic is refined not merely by inserting parentheses but also by tb* use of the small zeros. PAIRS OF QUADRATIC FORMS 309 The theorem just proved shows that pairs of homogeneous quadratic equations, of which the second equation in each pair is non-singular, may be classified by the use of their characteristics precisely as was done in the last section for pairs of quadratic forms. We proceed to illustrate this in the case w = 3, where we may con- sider that we have to deal with the classification of pairs of conies in a plane, one of the conies being non-singular. We have here three categories represented by the following normal forms : * I [1 1 11 i '^ ~ ^^"^^ "^ ^^2^ - \4 \ir= x\ + xl - 4 II. [2 1] \^='^\^i^-i + ^l + \^l lif-= 2x-^x^ + xl. III. r^l \<t>=^}^x^x^ + \ ^xl + 2x^x^ l-./r= 2x^X^ + xl. We next subdivide these categories into classes, and, by an ex- amination of the normal form in each case, we are enabled at once to characterize each class by certain projective properties which it has, and which are shared by no other class, f Since the conic i/r is non- singuhir in all cases, this fact need not be explicitly stated. [Ill] ^ and •^ intersect in four distinct points. [(1 1) 1] ^ and -^ have double contact. [(1 1 1)] <f} and •^ coincide. [2 1] <l> and •^ meet in three distinct points at one of which they touch. [(2 1)] ^ and -^ have contact of the third order. [3] ^ and i/r have contact of the second order. In all of the above cases (^, as well as ■y^, is non-singular. In the next five cases, <^ consists of a pair of distinct straight lines. * We assign to the constants c; and ^j, in formula (3) of the last section, values so chosen that the loci = 0, ^ = are real when the constants Xj are real. This is, of course, not essential, since we are not concerned with questions of reality. t In order to verify the statements made below, the reader should have some knowledge of the theory of the contact of conies; cf. for instance Salmon's Conia Sections, Chapter XIV., pages 232-238. alO INTRODUCTION TO HIGHER ALGEBRA [111] <f) and t/t intersect in four distinct points. [(1 1) 1] Both of the lines of which (f) consists touch ^jr. [2 1] One of the lines of which ^ consists touches -^^ while the other cuts it in two points distinct from the point of contact of the first. [2 1] The two lines of which consists intersect on ■\jr, and neither of them touches sjr. [3] The two lines of which A consists intersect on yjr, and one of them touches yfr. In the next two cases, <f) consists of a single line. [(11)1] The line cf> meets tjr in two distinct points. [(2 1)] The line <^ touches •^. Finally we have the case: » [(1 11]) Here </> = 0, and we have no conic other than i^. Suppose finally that we wish to classify not pairs of quadratic forms or equations but pencils of quadratic forms or equations. Consider the pencil of quadratic forms ^ _ ^ . / where ^ and yjr are quadratic forms, and yjr is non-singular, and suppose that the elementary divisors of the pair of forms <j>, yfr are given by formula (1) above. The question presents itself whether, if, in place of the forms <f>, y{r, we take any other two forms of the the constants fi, v being so chosen that jjl^^^v and that i|rj is non- singular, the pair of forms 0j, i^j, will have these same elementary divisors (1). If this were the case, we could properly speak of (1) as the elementary divisors of the pencil. This, however, is not the case, and the pencil of quadratic forms cannot properly he said to have elementary divisors.* * We here regard the pencil as merely an aggregate of an infinite number of quadratic forms, namely, all the forms which can be obtained from the expression <j> — Xfbj giving to \ different values. In this sense we cannot speak of the elementary divisors of the pencil. If, however, we wish to regard the polynomial in the k's and \, 4> — Xi/*, as the pencil, we may speak of its elementary divisors, meaning thereby simply what we have called the elementary divisors of the pair- of forms <t>,\l/.- PAIRS OF QUADRATIC FORMS 311 There is, however, a simple relation between the elementary divisors of two pairs of forms taken from the same pencil. In order to show this, let us determine the elementary divisors of the pair of forms ^j, i|rj, above. For this purpose consider the expression <pi — \i/rp which, when X =?fc 1, may be written (3) </,i-X^i=(l-X)[<^-X't] where \' = ^~ ^ . 1 — X Now suppose, as above, that X — a is any one of the linear factors of the matrix of ^ — X-v|r, and that l^ is the exponent of the highest power of X — « which is a factor of all the i-rowed determinants of this matrix. Then any one of the i-rowed determinants of the ma- trix oi (f>— \'^]r may, when X t^ 1, be written in the form (X' - affix') where / is a polynomial in X' of degree not greater than i — li. Accordingly, by (3), the corresponding i-rowed determinant of the matrix of <^j — Xyjr-^ may be written lfj,-j,-X-a{l-X)yif^{X) where /j is a polynomial in X. Thus we see that [-Bf]' is a factor of every i-rowed determinant of the matrix of ^j — Xt^Tj. Similar reasoning, carried through in the reverse order, shows that this is the highest power of a—jj, A. — a— V which is a factor of all these i-rowed determinants. Hence Theorem 2. If the pair of quadratic forms <f>, yjr, of which the second is non- singular, have the elementary divisors (X - Xi)% (X - \i)% (X - x,yK and if fJ; V are any two constants distinct from each other and such that V is distinct from all the constants X^, Ag, ••• A*, then the two forms (f)^ = (f>— /Ai/r, ^jf^ = 4>— vyjr, of which the second will then he non-singular, will have the elementary divisors (\_x;)% (X-X^)S (X-XifcX* where Xf=^ (i= 1, 2, ... A). 312 INTRODUCTION TO HIGHER ALGEBRA In particular, it will be seen that the two pairs of forms ^, y}r and <^j, i^j have the same characteristic [ej e^ ■■■ ej even if we put in parentheses to indicate which of the e's correspond to equal \/s. The characteristics will not, however, necessarily be the same if we put in small zeros to indicate which of the e's correspond to vanish- ing Vs» since \ and Xj do not usually vanish together. Accord- ingly, in classifying pencils of quadratic forms, we may use the characteristic of any pair of distinct forms of the pencil, the second of which is non-singular, but we must not introduce the small zeros into these characteristics. This classification, of course, applies only to what may be called non-singular pencils, that is, pencils whose forms are not all singular. It will readily be seen that what has just been said applies with- out essential change to the case of pencils of homogeneous quadratic equations. We may therefore illustrate it by the classification of non-singular pencils of conies.* We have here six classes of pencils which we characterize as follows : [1 1 1] The conies all pass through four distinct points. [(1 1) 1] The conies all pass through two points at which they have double contact with each other. [(1 1 1)] The conies all coincide. [2 1] The conies all pass through three points at one of which they touch one another. [(2 1)] The conies all pass throu'gh one point at which they have contact of the third order. [3] The conies all pass through two points, at one of which they have contact of the second order. EXERCISES 1. Determine, by the use of elementary divisors, the nature of each of the following pairs of conies : ■ 8xia;2 — 102:2X3 + 4xiXs = ■4xiX2— Q X2XS + G xixs = 0. (a) f3x5 + 7xi +i ^ \2xl + Zxl-xl + ' (6) { 3 x? — x| — 3 x§ — 3 xi X2 + 3 12 xs -I- xixs = 2 3:1 + xj— x^ — 2 xi X2 — 2 X2X3 + 2 xi X3 = 0. 2. Give a classification of pairs of binary quadratic equations, the second aquation of each pair being non-singular, and interpret the work geometrically. * For a similar classification of pencils of quadrios we refer to p. 46 of Bromwioh's book : Quadratic Forms and their Classification by Means of Invariant Factors. PAIRS OF QUADRATIC FORMS 313 106. Conclusion. We wish, in this section, to point out some of the important questions connected with the subject of elementary divisors, which, in order to keep our treatment within proper limits, we have been obliged to leave out of consideration. If 0j, i/rj and <j>2, y)r^ are two pairs of bilinear or quadratic forms of which t/t j, -^^ are non-singular, we have found a method of determining whether these two pairs of forms are equiva- lent or not. If we use the invariant factors instead of the ele- mentary divisors, our method involves only the use of the rational operations (addition, subtraction, multiplication, and division), and can, therefore, be actually carried through in any concrete case. In fact we have explained in § 93 some really practical methods of determining the invariant factors of a X-matrix, so that the problem of determining whether or not two pairs of bilinear or quadratic forms, the second form in each pair being non-singular, are equivalent, may be regarded as solved, not merely from the theoretical, but also from the practical point of view. There is, however, another question here, which we have not treated, namely, if the two pairs of forms turn out to be equiva- lent, to find a linear transformation which carries over one into the other. This problem, too, we may consider that we have solved from a theoretical point of view; for the proof we have given that if two pairs of forms have the same elementary divisors there exists a linear transformation which carries over one pair of forms into the other, consisted, as will be seen on examination, in actually giving a method whereby such a linear transformation could be determined. In fact, in the case of bilinear forms, the processes involved are, here again, merely the rational processes ; so that, given two equivalent pairs of bilinear forms, the second form of each pair being non-singular, we are in a position to find, in any concrete case, linear transformations of the a;'s and ^s which carry over one pair of forms into the other. Even here the arrangement of the work in a practical manner might require further consideration. In the case of quadratic forms the problem becomes a much more difficult one, inasmuch as the processes involved in the determination of the required linear transformation are no longer rational ; cf. the Lemma of § 101. That this is not merely a defect of the method we 314 INTRUDUCTIO.V TO HIGHER ALGEBRA have used, but is inherent in the problem itself, will be seen by a con- sideration of simple numerical examples. Let, for instance, (f)^ = 2x^ + 'Sxl, ^^ = 2x\—2>x\, iri= *i+ 2;| ■^i- ^ 2_ 1 Here the pairs of forms (^j, -^j and ^g, 1^2 'f^oth have the elementary divisors -> o ^ q A — ^, A — o, and are therefore equivalent. The linear transformation which carries over one pair of forms into the other cannot, however, be real (and therefore its coefBcients cannot be determined rationally from the coefficients of the given forms) since ^j and y^^ are definite, 02 and i/tj indefinite. We have, therefore, here the problem of devising a practical method of determining a linear transformation which carries over a first pair of quadratic forms into a second given equivalent pair. A method of this sort, which is a practical one when once the elemen- tary divisors have been determined, will be found in Bromwich's book on quadratic forms referred to in the footnote on p. 312. Another point at which our treatment is incomplete is in the restriction we have always made in assuming that, in the pair of bilinear or quadratic forms (^, i^, the form 1^ is non-singular. Al- though this is the case in many of the most important problems to which one wishes to apply the method of elementary divisors, it is still a restriction which it is desirable to remove. This may be done in part by making use not, as we have done, of the pencil ^ — \yjr, but of the more general pencil /x</) — \i/r, /j, and X being variable param- eters. The determinants of the matrix of this pencil are binary forms in (/u, \), and the whole subject of elementary divisors admits an easy extension to this case, the elementary divisors being now integral powers of linear binary forms. The only case which can- not be treated in this way is that in which not only (f) and i|r are both singular, but every form of the pencil /^(^ — X->fr is singular. This singular case, which was explicitly excluded by Weierstrass in his original paper, requires a special treatment which has been given by Kroneckef. Cf., for the case of quadratic forms, the book of Bromwich already referred to. Still another question is the application of the method of ele- mentary divisors to the case in which the two forms (p, i|r are real. PAIRS OP QUADRATIC FORMS 315 and only real linear transformations are admitted. In the case of bilinear forms, this question presents no serious difficulty ; cf. the exercises of §§ 97, 99. In the case of quadratic forms, however, the irrational processes involved in the proof of the Lemma of § 101 introduce an essential difficulty, since they are capable of introduc- ing imaginary quantities. Moreover, this difficulty does not lie merely in the method of treatment. The theorems themselves vyhich we have established do not remain true, as is seen by a refer- ence to the numerical example given earlier in this section for an- other purpose, where we have two pairs of real quadratic forms which, although they have the same elementary divisors, are not equivalent with regard to real linear transformations. We must content ourselves with merely mentioning this impor- tant subject, and referring, for one of the fundamental theorems, to p. 69 of the book of Bromwich. For further information concerning the subject of elementary divisors the reader is referred to Math's Theorie und Anwendung der Elementartheiler, Leipzig, Teubner, 1899. In English, the book of Bromwich already referred to and some sections in Mathews' revision of Scott's Determinants will be found useful. INDEX (The numbers refer to the pages.) Abelian group, 83. Adjoint of a determinaut, 30 ; of a matrix, 77, 80 ; of a quadratic form, 159. Affine transformation, 70. Algebraic complement, 23 Anharmonic ratio, 103. Associative law for matrices, 64. Augmented matrix, 44. Axis coordinates, 113. Bezont's method of elimination, 238. Bilinear forms, 114-117 ; determinant, matrix, rank of, 114; equivalence of, 116, 283 ; normal forms of, 116, 288, 289; pairsof, 283, 287-292; reducibility of, 116 ; singular, 114. Binary forms, 5; biquadratic, 260; cubic, 239; discriminant of, 237; factors of, 188 ; invariants of, 235; resultant of, 201, 236, 239; symmetric functions, 255. Biquadratic, binary, 260. Boole, 260. Bordered determinants, 28, 156-160. Bromwich, 270, 312-315. Cancellation, 7. Category of pairs of bilinear forms, 287 ; of collineations, 293; of pairs of quadratic fofms, 305. Cayley, 63, 154, 260, 296. Characteristic equation, function, matrix, 282. Characteristic of a quadratic form, 149 ; of a collineation, 293 ; of a pair of bilinear forms, 287 ; of a pair of quadratic forms, 305. Class of objects, 81 ; of quadratic forms, 148. Cofactor, 23. Cogredient variables, 90. Collineation, 68, 284, 292. Combinant, 115. Commutative group, 83; law for matrices, 63. Complement of a minor, 23. Complete system of invariants, 93. Complex quantity, 8, (iO. Component of complex quantity, 61. Composite elementary divisors, 270. Concomitants, 109. Cone, 120-123, 156. Conjugate of a matrix, 21, 65, 80; planes, 158 ; points, 121. Continuity, 14. Contragredient variables, 108. Contra variant, 109. Coordinates, homogeneous, 11 ; point, plane, line (axis, ray), 107-113. Correlation, 117. Corresponding polynomials, 178. Covariant, absolute, 91 ; integral rational, 99; relative, 97. Cramer's Rule, 43. Cross-ratio, 103, 107. Cubic, binary, 239. Cyclic group, 87. Definite quadratic form, 150. Degree of a polynomial, 1, 4; of a X-matrix, 277 ; of a product, 6, 277. Descriptive property, 88, 232. Determinant, 20; adjoint of, 30; bordered, 28, 156-160; Laplace's development of, 26; matrix of, 21 ; minors of, 22 ; of a bilinear form, 114 ; of a matrix, 21 ; of a transformation, 66 ; orthogonal, 154 ; 317 318 INDEX Determinant — Continued. product of two, 26 ; rank of, 22 ; skew-symmetric, 59 ; symmetric, 5t). Discriminant of a binary biquadratic, 260; of a binary cubic, 239; of a binary form, 237, 259; of a polynomial in one variable, 2JiO; of a quadratic form, 128; of a quadric surface, 118. Division of polynomials, 180 ; of X-matrices, 278 ; of matrices, 75. Divisor of zero, 65, 80. Domain of rationality, 175, 212, 216 Dyad (dyadic polynomial), 79. Dyalitic method of elimination, 199. Element of a determinant or matrix, 20; of a set, system, or group, 81. Elementary divisors of a X-matrix, 271 ; of a ?ollineation, 286; of a pair of bilinear forms, 284; of a pair of quadratic forms, 303 ; simple, composite, 270. Elementary symmetric function, 242, 253. Elementary transformation of a matrix, 55; of a X-matrix, 262. Elimination, 198, 217, 238. Equations, linear, 43 ; homogeneous, linear, 47 ; quadratic, 149; quadratic, pairs of, 307; quadratic, pencils of, 312. Equianharmonic points, 107. Equivalence, 92 Equivalent matrices, 55, 93; collineations, 286; X-matrices, 263, 274 ; pairs of bilinear forms, 283; pairs of matrices, 281 ; pairs of quadratic forms, 170, 303; quadratic forms, 135, 148. Euclid's algorithm, 189, 192, 206. Euler's theorem for homogeneous functions, 237. Factors of a polynomial, 174, 187, 203; see invariant factors. Fixed points of a collineation, 285 ; lines, planes of a collineation, 286. Forms, 4; see bilinear, quadratic, binari/ forms ; biquadratic, 260: cubic, 239; polar, 127. Fours group, 87. Fractional matrices, 86. Frobenius, 2(i2, 270. Fundamental system of solutions, 49; theorem of algebra, 16. Generator of a quadric surface, 119. Gibbs, 79. Greatest common divisor of integers, 188 ; of polynomials in one variable, 191, 197, of polynomials in two variables, 206. Ground-form, 96. Group, 80; Abelian or commutative, 83; cyclic, 87 ; fours group, 87 ; isomorphic, 83 ; sub-, 83. Group property, 82. Hamilton, 79, 296. Harmonic division, 104. Homogeneity, principle of, 226. Homogeneous coiirdinates, 11 ; invariants, 2.30; linear equations, 47, 49; polynomials, 4. Idemfactor, 74. Identical vanishing (equality) of polynomi. als, 2, 5, 7, 10; element of a group, 82 ; transformation, 67. Hidefinite quadratic form, 150. Index of inertia of quadratic form, 146. Invariant, absolute algebraic, 89; arithmetical, 91, 94, 115, 124, 129, 146, 2ST ; complete system of, 93 ; geometric, 88, 103; homogeneous, 230; integral rational, 99, 101, 115, 129, 137, 159, 166, 218, 2.39, 260; irrational, 167, 259, 260 ; rational, 96, 222; see also integral, ni tionat; relitive algebraic, 96, 115; see integral rational. Invariant factors, of a X-matrix, 269; of a collineation, 286; of a pair of bilinear forms, 284 ; of a pair of quadratic forms, 303. Inverse of a transformation, 67 ; of an element of a group, 82; of a matrix, 75, 80; of a quadratic form, 160. Isobaric polynomial, 222; symmetric function, 245, 255, 256. Isomorphic groups, 83. INDEX 319 Jacobi, 144. Kronecker, 4, 139, 262, 314. X-equation of two conies, 164 ; of two quadratic forms, 166. X-matrix, 2()2. Lagraiige's reduction, 131. Laplace's development, 26. Law of Inertia, 144 ; of Nullity, 78, 80. Line at infinity, 13. Line-coordinates, 108, 110. Linear dependence, conditions for, 36-38 ; of geometric configurations, 39; of polynomials, 35, 38 ; of sets of constants, 35, 48. Linear equations, 43, 47, 49; transformations, 66. Linear factors of polynomials in one vari- able, 187 ; of binary forms, 188; of X-matrices, 270. Matrix, theory of, 20-22, 51-66, 74-80, 86, 93, 262-283, 296-302 ; adjoint of, 77, 80 ; as a complex quantity, 60 ; augmented, 44; conjugate, 21 ; determinant of, 21 ; division of one by another, 75 ; elementary transformation of, 55; equivalent, 55 ; fractional, 86 ; inverse of, 75 ; multiplication by matrix, 63; multiplication by scalar, 62 ; normal form of a, 56 (Exercise 3) ; normal form of a symmetrical, 59; o' a bilinear form, 114 ; of a determinant, 21 ; of a quadratic form, 128; of a quadric surface, 118 ; ' of a system of linear equations, 44; of a transformation, 66. orthogonal, 154, 302, 304 ; powers of, 75 ; product of two, 63 ; rank of, 22 ; rank of product of two, 77 ; scalar, 76 ; similar, 283; singular, 65 ; skew-symmetric, 59; sum or difference of two, 62 ; Matrix — Continued. , symmetric, 56 ; transposed, 21; unimodular, 83 ; unit, 74. Minors of a determinant, 22 ; complementary, 22 ; corresponding, 31; principal, 23, 57-59. Mixed concomitants, 109. Moore, 86. Multiplication theorem, 28. Multiplicity of roots of an equation, 18; of pieces of curves and surfaces, 211, 211 Neighborhood of a point, 8, 16, 214. Newton's formulae, 244. Normal form of a bilinear form, 116; of a binary biquadratic, 261 ; of a binary cubic, 239; of a X-matrix, 267 ; of a matrix, 56 (Exercise 3) ; • of a pair of bilinear forms, 289 ; of a pair of quadratic forms, 169, 171, 306; of a quadratic form, 135 ; of a quadric surface, 124; of a real quadratic form, 148; of a symmetrical matrix, 59. Nullity, Sylvester's Law of, 78, 80. Null-system, 117. Order of a determinant or matrix, 20 ; of a group, 87. Orthogonal transformation, matrix, determi nant, 154, 173, 302, 304. Pencil of conies, 163, 312; of bilinear forms, 279 ; of quadratic forms, 165, 310. Period of an element of a group, 87. Plane at infinity, 13,; conjugate, 158. Plane-coordinates, 107. Point in space of /i dimensions, 9; at infinity, 12; conjugate, 121; equation of a, 107, 108 ; neighborhood of a, 8, 16, 214. Polar plane, 122; form, 127; tetrahedron, 125. Pole, 124. Polynomial, definition, degree of, etc., 1, 4; continuity of a, 14; corresponding, 178; dyadic, 79 ; 820 INDEX Polynomial — Continued. in a matrix, 296 ; isobaric, 2^2; linear dependence of, 35 ; real, 5; roots of a, 18 ; symmetric, 240, 252. Prime (relatively) polynomials, 175. Principal minors, 23, 57-50. Product, degree of, 1, 4; of determinants, 26; of matrices, 63, 277. Projective transformation, 69; property, 88, 232. Pseudo- tangent lines and planes, 120. Quadratic forms, 127 ; adjoint of, 15i) ; definite and indefinite, 150; invariants of, 129, 137, 146, 159, 166, 303; inverse or reciprocal of, 160; law of inertia of, 114; matrix, discriminant, rank of, 128; normal forms of, 135, 148, 169, 171, 306 ; pairs of, 165, 302 ; polar of, 127 ; real, 144 ; reducibility of, 136, 147 ; reduction of, to sum of squares, 131, 139, 167, 170, 173 ; regularly arranged, 147 ; signature of, 146 ; singular, 128; vertex of, 129. Quadric surface, matrix, discriminant, rank of, 118; classification of, 123, 149, 173 ; ruling of, 119; singular, 118; tangent to, 119, 120, 155. Quantic, 4; Quaternary form, 5. Kank of a matrix or determinant, 22, 54-59; of a bilinear form, 114; of a \-matrix, 262 ; of a quadratic form , 128 ; of a quadric surface, 118 ; of a system of homogeneous linear equa- tions, 47 ; of a system of points or linear forms, 94; of the adjoint of a quadratic form, 161 ; of the product of two matrices, 77. Rational invariants, 96, 222; see also in- variant, integral, rational ; of a X-matrix, 270. Rational relation, 244. Rationality, domain of, 175, 212, 216. Ray coordinates, 113. Real polynomials, 5, 174 ; matrix, X-matrix, elementary transforma- tion, 278 ; quadratic forms, 144-154, 161, 170-173. Reciprocal or inverse of a quadratic form, 160. Reciprocation, 117. Reducibility of a polynomial, 174 ; in a domain, 174, 175 ; of bilinear forms, 116 ; of binary forms, 188; of determinants, 176; of polynomials in one variable, 187; of quadratic forms, 136, 147. Regularly arranged quadratic form, 147. Resultant of linear forms, 95; of two binary forms, 201, 236, 239, 257 ; of two polynomials in one variable, 195, 239, 248. Roots of a polynomial or equation, 18. Ruling of a quadric surface, 119. S-functions, 241, 253. S-functions, 240, 252. Scalar, 62 ; matrix, 76. Self-conjugate tetrahedron, 125; triangle, 164. Semi-defluite quadratic form, 150. Set of objects, 80. Sgn, 147. Signature of a quadratic form, 146. Similar matrices, 283. Simple elementary divisors, 270. Singular matrix, 65; bilinear form, 114; conic, 163, 272; linear transformation, 67 ; quadratic form, 128; quadric surface, 118. Skew-symmetric determinant, 59; bilinear form, 117 ; matrix, 59. Smith, H. J. S., 262. Subgroup, 83. Subresultant, 197. Sylvester, 78, 144, 199, 262. Symbolic product of bilinear forms, 114. Symmetric determinant and matrix, 50 299; bilinear form , 115 ; binary function, 255; polynomial, 240; pAjynomial in pairs of variables, 252: ternary function, 257. System, 80. INDEX 821 Tangent lines and planes to quadric surface, true and pseudo, 119, 120. Ternary form, 5 ; symmetric fiuiction, 257. Transformation, affiue, 70 ; determinant and matrix of, 66; elementary (of a matrix), 55, 262; identical, 67; inverse, 67 ; linear, 66 ; orthogonal, 154, 173, 304 ; projective, 69 ; singular, 67. Transposed matrix, 21. Unimodular matrix, 83. Unit matrix, 74. Vertex of a cone, 120, 122, 123, 156; of a quadratic form, 129. Weierstrass, 262, 270, 314. Weight of an invariant, 96, 225; of a covariant, 97, 226 ; of a polynomial, 222 ; of a symmetric polynomial, 245, 253. Cornell University Library QA 155.B66 Introduction to higher algebra. 3 1924 002 750 473 Digitized by Microsoft®