Spectral density of the non-central correlated Wishart ensembles

TL;DR

This paper derives a self-consistent equation for the spectral density of non-central correlated Wishart matrices, extending the understanding of their eigenvalue distributions in large dimensions.

Contribution

It introduces the non-central correlated Wishart ensemble and provides a theoretical framework to analyze its spectral density using the Pastur equation.

Findings

Derived the Pastur self-consistent equation for nc-CWE

Characterized the spectral density at large matrix dimensions

Extended Wishart ensemble analysis to non-zero mean matrices

Abstract

Wishart ensembles of random matrix theory have been useful in modeling positive definite matrices encountered in classical and quantum chaotic systems. We consider nonzero means for the entries of the constituting matrix A which defines the correlated Wishart matrix as W = AA{\dag}, and refer to the ensemble of such Wishart matrices as the non-central correlated Wishart ensemble (nc-CWE). We derive the Pastur self-consistent equation which describes the spectral density of nc-CWE at large matrix dimension.