146 THE FACTORS OF THE MIND

other influences. No doubt, the isolation will never be
perfect, and the independence will always be approximate
and never complete : but limitations of this sort pervade
even the simplest investigations of physics and mechanics.
Accordingly, let us study the consequences of this artificial
isolation.

The Single-factor Theorem.—If any table of selected
measurements were due exclusively to the operation of one
factor, our initial equation would be reduced to an equation
with one term only on the right-hand side, namely,
ttfy. = figg.. That means that, with n tests, we should have

mu : m* : . . . : mni = flg: /2ff: . . . : fng (i — l,29...N)
for each of the N individuals. Thus, corresponding pairs
of measurements would always stand in exactly the same
proportion, and, to borrow a convenient term from matrix
algebra, the whole table of measurements would form a
matrix of £ rank' one.1 The converse of this is the single-
factor theorem for a measurement matrix : " any n X N
table, in which the rows (and therefore also the columns)

1 This conception seems to me so important that a concrete illustration
may be given for the benefit of the non-mathematical reader. Suppose, as
many Art Schools used to teach, that all the measurements of the normal
human body ought ideally to bear the same characteristic ratio to the total
height, whatever the individual's height might actually be (e.g. head-height
•J, trunk-length ^(y, leg-length -J, arm-length J, shoulder-breadth \> waist-
breadth -J, of the total height); then, on measuring N persons of varying
stature, all the figures should still be in constant proportion, whatever the
individual's actual height. If we merely know the height of the N individuals
—65, 66, 67, ... inches, say,—we could deduce the probable lengths of their
limbs, etc,, by multiplying these figures by the column of fractions; and the
n X N table of physical measurements so obtained would have a * rank' of
one.

For examples of matrices of rank one in hypothetical frequency tables
filled in on the assumption of homogeneity, ci [25], p. 66, Table 111
and [50], p. 91, Table 18,

The simplest definition of rank is the following. If every row (or column)
in a matrix can be expressed as a linear combination of r linearly independent
rows only (but no less), then the matrix has a rank of r. From this, the reader
with an elementary knowledge of determinants will easily derive the more
usual formal definition : a matrix is of rank r when at least one of its minor
determinants of order r does not vanish, while all its minor determinants of
order (r + i) do vanish. Thus, the criterion for a matrix of rank one is that
all its two-rowed minors must be zero.