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

Abstract

Suppose that A is an n× n positive definite Hermitian matrix. Let X and Y be n× p and n× q matrices, respectively, such that X * Y= 0. The present article proves the following inequality, X ∗ AY Y ∗ AY − Y ∗ AX ⩽ λ 1−λ n λ 1+λ n 2 X * AX , where λ 1 and λ n are respectively the largest and smallest eigenvalues of A, and M − stands for a generalized inverse of M. This inequality 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 inequalities about covariance matrix and various correlation coefficients including the canonical correlation, multiple and simple correlations. Some applications in parameter estimation are also given.