ABSTRACT
The left-hand side of (11.5) is a polynomial of degree p in ? and hence has p roots (say) ?1=···=?p and ?=a'S(12)ß is the correlation between U1 and V1 subject to the restriction (11.1). From (11.4-11.5) we get
which has p1 solutions for ? 2, (say), and p1 solutions for a, and
which has p2 solutions for ? 2 and p2 solutions for ß. Now (11.6) implies that
Since
we conclude that (11.6) and (11.7) have the same solutions. Thus (11.5) has p roots of which p2-p1 are zeros, and the remaining 2p1 nonzero roots are of the form ?=±pi, i=1,…, p1. The ordered p roots of (11.5) are thus
To get the maximum correlation of U1, V1 we take ?=?1. Let a (1), ß(1) be the
solution (11.4) when ?=?1. Thus U1=a (1)'X(1), V1=ß
(1)'X(2) are normalized (with respect to variance) linear combinations of X(1), X(2), respectively, with maximum correlation ?1.
