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

Let W be an n×nn×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 (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−nm−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(⋅)tr(⋅) denotes the trace of a square matrix. Moreover, we establish that, as m,n→∞m,n→∞ with m−nm−n fixed, tr(W)/λmin scales like n3n3. Finally, we find the average of the reciprocal of the characteristic polynomial det[zIn+W],|argz|<π, where In and det[⋅]det[⋅] denote the identity matrix of size nn and the determinant, respectively.