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

✍ Scribed by Song-Gui Wang; Wai-Cheung Ip

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
110 KB
Volume
296
Category
Article
ISSN
0024-3795

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✦ Synopsis

Suppose that A is an n  n positive de®nite Hermitian matrix. Let X and Y be n  p and n  q matrices, respectively, such that X à Y 0. The present article proves the following inequality,

where k 1 and k n are respectively the largest and smallest eigenvalues of A, and M À 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 coecients including the canonical correlation, multiple and simple correlations. Some applications in parameter estimation are also given.

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