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

Let W be an n × n complex non-central Wishart matrix with m ( ź 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 ( λ 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 ) / λ min , where tr ( ź ) denotes the trace of a square matrix. Moreover, we establish that, as m , n ź ∞ with m - n fixed, tr ( W ) / λ min scales like n 3 . Finally, we find the average of the reciprocal of the characteristic polynomial det z I n + W , | arg z | < π , where I n and det ź denote the identity matrix of size n and the determinant, respectively.