2013/06/27 by Dharmawansa, Prathapasinghe
#15B2 #60B20 #62H10 #FOS: Mathematics #G.1.0 #G.1.3 #G.3 #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.1306.6566
Recently, D. Wang has devised a new contour integral based method to simplify certain matrix integrals. Capitalizing on that approach, we derive a new expression for the probability density function (p.d.f.) of the joint eigenvalues of a complex non-central Wishart matrix with a rank-1 mean. The resulting functional form in turn enables us to use powerful classical orthogonal polynomial techniques in solving three problems related to the non-central Wishart matrix. To be specific, for an n× n complex non-central Wishart matrix W with m degrees of freedom (m≥ n) and a rank-1 mean, we derive a new expression for the cumulative distribution function (c.d.f.) of the minimum eigenvalue (λmin). 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 for the minimum eigenvalue which takes the form of the determinant of a square matrix of size m-n with the Bessel kernel. We also develop a moment generating function based approach to derive the p.d.f. of the random variable \fractr(W)λmin, where tr(⋅) denotes the trace of a square matrix. This random quantity is of great importance in the so-called smoothed analysis of Demmel condition number. Finally, we find the average of the reciprocal of the characteristic polynomial det[zIn+W], |arg z|