Three Problems Related to the Eigenvalues of Complex Non-central Wishart Matrices with a Rank-1 Mean

Abstract

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 $n\times n$ complex non-central Wishart matrix $\mathbf{W}$ with $m$ degrees of freedom ($m\ge n$) and a rank-1 mean, we derive a new expression for the cumulative distribution function (c.d.f.) of the minimum eigenvalue (${\lambda }_{\min }$). 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 $\frac{\text{tr}(\mathbf{W})}{{\lambda }_{\min }}$, where $\text{tr}(\cdot )$ 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 $\det [z{\mathbf{I}}_n+\mathbf{W}],|\arg z|<\pi$, where ${\mathbf{I}}_n$ and $\det [\cdot ]$ denote the identity matrix of size $n$ and the determinant, respectively.

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• Hypergeometric Functions of Matrix Arguments and Linear Statistics of Multi-Spiked Hermitian Matrix Models
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