264 THE FACTORS OF THE MIND

who prefer the language of c analysis of variance/ we may describe
our procedure as seeking first the factor whose contribution to the
total variance shall be as large as possible : then the factor whose
contribution to the remaining variance shall be as large as possible ;
and so on : until the ultimate remainder is zero or rather statistically
negligible as judged by the probable error.

It is easy to show that these requirements are equivalent
to reducing the correlation matrix R to a diagonal canonical
form by an orthogonal transformation. We obtain
L'fLL = V ; so that we can write F = LV1, V denoting the
diagonal matrix of latent roots (or factor- variances), and L
the orthogonal (or semi-orthogonal) matrix of latent vectors
(or direction cosines) ([93], p. 290 ; [102], p. 177).

(iv) Factors defined by Selective Operators. — The fore-
going procedure brings the analysis into line with that
adopted for analogous problems in other sciences. We
have Mk = l&M = EkM, where Ek = U/, is a unit
hierarchy, and E19 E2, . . ., Ek9 ... form a * spectral set of
selective operators.' l Thus, we can carry our previous

1 In its simplest application the notion of a selective operator may be
explained as follows. Let M be a mixed population consisting of r mutually
exclusive classes or types, say, Europeans (M^9 Indians (M2), and Chinese
(Af3). Let Ex be the selective operator which sorts out only the Europeans ;
E% the selective operator which sorts out only the Indians ; and so on. Then

but E^M + E^M + . . . + ErM= MI + M2 + . . . + Mr = M ;

i.e.27£ft=i ...... (i)

Again E^EJM = E^ = M ;

hence E£=Ek . . . . . (2)

(Le. Ek is c idempotent 5) : for, if we try to select the Europeans from a
selected group containing nothing but Europeans, we reach the same group
as before.
Once more EJE^M. = E2MI = o ; hence E^ = o, and generally

i#,= o; (/=M) ....... (3)

for if we try to select the Chinese from a selected group that contains no
Chinese, we obtain an empty class.

Operators possessing these three properties are said to constitute a spectral
set. (The term £ spectral ' is derived from the use of such operators in
atomic physics where on the basis of a spectral analysis of a mixed radiation
an attempt is made to segregate * pure components ' along the lines of the
Classical experiment of Stern ancj Qerlach.)