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

作     者：WANG Songgui Wai-Cheung IpDepartment of Applied Mathematics, Beijing Polytechnic University, Beijing 100022, China Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100080, China Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, China 

作者机构：Department of Applied Mathematics Beijing Polytechntc University Beijing China Institute of Applied Mathematics Chinese Academy of Sciences Beijing China Department of Applied Mathematics Hong Kong Polytechnic University Hong Kong China 

出 版 物：《Chinese Science Bulletin》 (科学通报(英文版))

年 卷 期：1999年第44卷第2期

页      面：118-121页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

主　　题：Wielandt inequality Cauchy-Schwarz inequality Wishart matrix. 

摘      要：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（YAY）-Y*AX≤（（λ1-λn）/（λ1+λn）2）X*AX,where λ1 and λn 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 correlation.