Eigenvalue Density of Correlated Complex Random Wishart Matrices

TL;DR

This paper derives an exact formula for the eigenvalue density of correlated complex Wishart matrices, which are important in various scientific fields, using a novel character expansion method.

Contribution

It introduces a new exact calculation method for the eigenvalue density of correlated Wishart matrices with arbitrary positive definite matrices A and B.

Findings

Exact eigenvalue density formula derived

Applicable to matrices of arbitrary dimensions

Method useful for multivariate analysis and information theory

Abstract

Using a character expansion method, we calculate exactly the eigenvalue density of random matrices of the form M^\dagger M where M is a complex matrix drawn from a normalized distribution P(M) ~ exp(-\Tr(A M B M^\dagger) with A and B positive definite (square) matrices of arbitrary dimensions. Such so-called ``correlated Wishart matrices'' occur in many fields ranging from information theory to multivariate analysis.