APPENDIX II

ANALYSIS OF A MATRIX INTO ITS LATENT ROOTS
AND LATENT VECTORS

Tine ultimate result of analysing a correlation table by the method
of ' least squares ' or * weighted summation ' is to reduce the initial
matrix of measurements to terms of the latent roots and latent
vectors of the correlation matrix. It can thus be expressed as the
product of three component matrices—a diagonal matrix, pre-
multiplied and post-multiplied by two semi-orthogonal matrices, or,
in matrix notation, us M « ZjKP, where L is the ' modal matrix'
of' latent vectors/ V the diagonal matrix off latent roots' or factor
variances, and P the semi-orthogonal( population' matrix of factor-
measurements. Where it is necessary to distinguish matrices
obtained by correlating tests and persons respectively, a subscript
t or f will be affixed*

As I have vStated in the text, this mode of analysis seems to me the
most logical and the most useful; and, since the nomenclature of
matrix algebra may be unfamiliar, the following tables are appended
to illustrate the simplicity of the procedure. Incidentally they
will serve to demonstrate that, provided the initial matrix is suit-
ably standardized, the resulting * factors* are the same, whether
we begin by covariating persons or tests.

In order that the reader can follow the working mentally, I have
taken a small fictitious table of integers, and have calculated
covariances, instead of reducing the figures to standard measure
and so calculating correlations* If desired, the correlations and the
factor-saturations1 could at once be obtained by dividing the
covariances and the factor-loadings by the square roots of the
variances of the initial measurements: e.g. in Table II, by <\/$6,
V'fco, and -\/2o.

The initial matrix, M, is set out at the head of Table I, and is
supposed to give the marks of 4 persons (pl9 f^ p& p4) in three

* These will not represent th© same set of factors as would have been obtained
by analysing the correlations directly. If we required factor-saturations directly
deduced from the corrections rather than covariances, we should have to start by
normalising th© Initial measurements at the outset, That would involve working
with decimal fractions running into several figures, and would be too complicated to
follow mentally*

4*7