Products of Complex Rectangular and Hermitian Random Matrices

TL;DR

This paper investigates the spectral properties of products of complex rectangular and Hermitian random matrices, deriving joint eigenvalue densities and transformation formulas, thus extending analytical techniques in random matrix theory.

Contribution

It introduces a novel approach to analyze products of complex and Hermitian matrices, generalizing existing methods for GUE and avoiding non-compact group integrals.

Findings

Derived joint probability density functions of eigenvalues.

Proved transformation formulas for bi-orthogonal functions.

Extended analytical techniques to broader classes of matrix products.

Abstract

Products and sums of random matrices have seen a rapid development in the past decade due to various analytical techniques available. Two of these are the harmonic analysis approach and the concept of polynomial ensembles. Very recently, it has been shown for products of real matrices with anti-symmetric matrices of even dimension that the traditional harmonic analysis on matrix groups developed by Harish-Chandra et al. needs to be modified when considering the group action on general symmetric spaces of matrices. In the present work, we consider the product of complex random matrices with Hermitian matrices, in particular the former can be also rectangular while the latter has not to be positive definite and is considered as a fixed matrix as well as a random matrix. This generalises an approach for products involving the Gaussian unitary ensemble (GUE) and circumvents the use there of non-compact group integrals. We derive the joint probability density function of the real eigenvalues and, additionally, prove transformation formulas for the bi-orthogonal functions and kernels.