LATENT ROOTS AND VECTORS

491

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Analysis by Selective Operators: p. 494. (a) Canonical Expansion
ofCovarianfff Matrix (sec p. 164).—The method of multiple-factor
analysis described in the text is based on the assumption that any
symmetric covariance (or correlation) matrix can be expressed as
the sum of a number of perfect hierarchies, each of which consists
of a * unit hierarchy' (Ej) multiplied by the corresponding factor-
variance (0y), Such a series I call the e canonical expansion ' of the
covariance (or correlation) matrix. A unit hierarchy is formed by
post-multiplying a latent vector (one column of Z/) by its transpose
(the same figures written as a row), according to the usual method
of constructing a theoretical hierarchy. Thus, in Table IVa, El is
formed from the first column of Lt (shown in Table 114), £2 from the
second column, and I£% from the otiose column required to turn the
semi-orthogonal Lt matrix into a completely orthogonal square
matrix. From the weighted sum of these unit hierarchies we can
reconstruct the original covariance matrix, R(, as shown in the
table* Each of the unit hierarchies represents the effect of a single
pure and independent factor; and the corresponding weighted
hierarchy represents the amount of correlation or covariance
attributable to that factor.

(i) Factorial Expansion of Measurement Matrix (see pp. 261—5).—
We can determine the contribution of such a factor to the empirical
matrix of test-measurements by simply pre-multiplying that
matrix by the appropriate unit hierarchy (see Table IVi). With
the multipliers calculated in this way, the result is to yield, with
EH the best approximation to M that is obtainable with a single
factor only, namely, Ml = E^M ; with £2, the best approximation
to the residuals, (M — Ma), that is obtainable with a single factor
only ; and so on (the closeness of the approximation being judged
by the principle of least squares), Thus, if we assume that there
can be only one true factor determining the initial test-measure-
ments, and that the remainder consists of errors of measurement,
then M% gives the best estimate of the hypothetical values. And the
same result will be readied whether we start by correlating rows or