Some new results on the eigenvalues of complex non-central Wishart matrices with a rank-1 mean

JOURNAL OF MULTIVARIATE ANALYSIS

卷:149

Article

Dharmawansa, Prathapasinghe^1,2 

[1] Univ Moratuwa, Dept Elect & Telecommun Engn, Moratuwa, Sri Lanka

[2] Stanford Univ, Dept Stat, Sequoia Hall,390 Serra Mall, Stanford, CA 94305 USA

关键词: Demmel condition number;    Eigenvalues;    Hypergeometric function of two matrix arguments;    Non-central Wishart distribution;    Random matrix;   

DOI  :  10.1016/j.jmva.2016.03.003

来源: Elsevier

【 摘 要 】

Let W be an n x n complex non-central Wishart matrix with m (>= n) degrees of freedom and a rank-1 mean. In this paper, we consider three problems related to the eigenvalues of W. To be specific, we derive a new expression for the cumulative distribution function (c.d.f.) of the minimum eigenvalue (lambda(min)) of W. The c.d.f. is expressed as the determinant of a square matrix, the size of which depends only on the difference m - n. This further facilitates the analysis of the microscopic limit of the minimum eigenvalue. The microscopic limit takes the form of the determinant of a square matrix with its entries expressed in terms of the modified Bessel functions of the first kind. We also develop a moment generating function based approach to derive the probability density function of the random variable tr(W)/lambda(min), where tr(.) denotes the trace of a square matrix. Moreover, we establish that, as m, n -> infinity with m - n fixed, tr(W)/lambda(min) scales like n(3). Finally, we find the average of the reciprocal of the characteristic polynomial det[zI(n) + W], vertical bar arg z vertical bar < pi, where I-n and det[.] denote the identity matrix of size n and the determinant, respectively. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

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