The canonical correlations of a 2×2 block matrix with given eigenvalues Original Research Article

Let 1less-than-or-equals, slantkless-than-or-equals, slantn/2, and image be an n×n positive definite matrix so that A11 is k×k. Suppose that A has given eigenvalues λ1greater-or-equal, slantedcdots, three dots, centeredgreater-or-equal, slantedλn>0. The singular values σj(A11−1/2A12A22−1/2) (j=1,…,k) are known as the canonical correlations of the partitioned matrix A and have been extensively studied with regard to the inefficiency of the ordinary least squares method in statistics. The object of this paper is to provide proofs of some new inequalities for the canonical correlations in terms of λ1,…,λn.