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

Abstract

LetWbe an n×n complex non-centralWishart 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 (λmin) 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)/λmin, where tr(·) denotes the trace of a square matrix. Moreover, we establish that, as m, n → ∞ with m − n fixed, tr(W)/λmin scales like n3. Finally, we find the average of the reciprocal of the characteristic polynomial det[zIn +W], | arg z| < π, where In and det[·] denote the identity matrix of size n and the determinant, respectively.