A matrix version of the Wielandt inequality and its applications to statistics

Suppose that A is an n x n positive definite Hermitian matrix. Let X and Y be n x p and n x q matrices, respectively, such that X*Y = 0. The present a rticle proves the following inequality, X*AY(Y*AY)Y-*AX less than or equal to (lambda(1)-lambda(n)/lambda(1)+lambda (n))X-2*AX, where lambda(1) and lambda(n) are respectively the largest and smallest eig envalues of A, and M- stands for a generalized inverse of M. This inequalit y is an extension of the well-known Wielandt inequality in which both X and Y are vectors. The inequality is utilized to obtain some interesting inequ alities about covariance matrix and various correlation coefficients includ ing the canonical correlation, multiple and simple correlations. Some appli cations in parameter estimation are also given. (C) 1999 Elsevier Science I nc. All rights reserved.