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

Authors:

Institution:

(1) Department of Applied Mathematics, Beijing Polytechntc University, 100022 Beijing, China;(2) Institute of Applied Mathematics, Chinese Academy of Sciences, 100080 Beijing, China;(3) Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, China

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.

Keywords:

Wielandt inequality Cauchy-Schwarz inequality Wishart matrix

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