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

Abstract

Suppose thatA is ann ×n positive definite Hemitain matrix. LetX andY ben ×p andn ×q matrices (p + q≤ n), such thatX* Y = 0. The following inequality is proved

$$X^* AY(Y^* AY)^ - Y^* AX \leqslant \left( {\frac{{\lambda _1 - \lambda _n }}{{\lambda _1 + \lambda _n }}} \right)^2 X^* AX,$$

where λ₁, and λ_n, are respectively the largest and smallest eigenvalues ofA, andM ^- stands for a generalized inverse ofM. This inequality is an extension of the well-known Wielandt inequality in which bothX andY 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.