THE EXACT DISTRIBUTION OF THE CONDITION NUMBER OF A GAUSSIAN MATRIX

被引：8

作者：

Anderson, William ^{[1
]}

Wells, Martin T. ^{[2
]}

机构：

[1] Cornell Univ, Sch Operat Res & Informat Engn, Ithaca, NY 14853 USA

[2] Cornell Univ, Dept Stat Sci, Ithaca, NY 14853 USA

来源：

SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS | 2009年 / 31卷 / 03期

关键词：

condition number; eigenvalues; random matrices; singular values; Wishart distribution;

D O I：

10.1137/070698932

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

Suppose G(pxn) is a real random matrix whose elements are independent and identically distributed standard normal random variables. Let W-pxp = G(pxn)G(nxp)(T), which is the usual Wishart matrix. In addition, let lambda(1) > lambda(2) > ... > lambda(p) > 0 and sigma(1) > sigma(2) > ... > sigma(p) > 0 denote the distinct eigenvalues of the matrix Wpxp and singular values of G(pxn), respectively. The 2-norm condition number of G(pxn) is kappa(2)(G(pxn)) = root lambda(1)/lambda(p) = sigma(1)/sigma(p), the square root of the ratio of largest to smallest eigenvalues of the Wishart matrix. In this article we derive an exact expression, albeit somewhat complex, for the probability density function of kappa(2)(G(pxn)).

引用

页码：1125 / 1130

页数：6

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