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

TL;DR

This paper derives an explicit formula for the density of complex eigenvalues of random matrices formed by a fixed positive semi-definite diagonal matrix multiplied by a Haar-distributed unitary matrix, advancing understanding of eigenvalue distributions.

Contribution

It provides a new explicit formula for the eigenvalue density of matrices with prescribed singular values and Haar-distributed unitary matrices.

Findings

Explicit eigenvalue density formula derived

Applicable to matrices with prescribed singular values

Enhances understanding of eigenvalue distributions in random matrix ensembles

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.