On the mean density of complex eigenvalues for an ensemble of random matrices with prescribed singular values

Abstract

Given any fixed $N \times N$ positive semi-definite diagonal matrix $G\ge 0$ we derive the explicit formula for the density of complex eigenvalues for random matrices $A$ of the form $A=U\sqrt{G}$} where the random unitary matrices $U$ are distributed on the group $\mathrm{U(N)}$ according to the Haar measure.