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

Suppose that A is an n×n positive definite Hemitain matrix. Let X and Y ben×p and n×q matrices(p+q≤n), such that X~*Y=O. The following inequality is provedX~*AY(Y~AY)~-Y~*AX≤((λ-λ)/(λ+λ)~2)X~*AX,where λand λare 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 correlation.##属性不符