Eigenvalues and Singular Values of Products of Rectangular Gaussian Random Matrices (The Extended Version)

TL;DR

This paper derives the spectral densities of eigenvalues and singular values for products of rectangular Gaussian random matrices using advanced mathematical techniques, providing new insights into their behavior in the large-size limit.

Contribution

It introduces polynomial equations for the M-transforms of spectral densities and proposes a conjecture linking eigenvalues and singular values in non-Hermitian matrices with rotational symmetry.

Findings

Derived mean spectral densities in the thermodynamic limit

Identified singular behavior near zero eigenvalues

Proposed a finite-size spectral density approximation

Abstract

We consider a product of an arbitrary number of independent rectangular Gaussian random matrices. We derive the mean densities of its eigenvalues and singular values in the thermodynamic limit, eventually verified numerically. These densities are encoded in the form of the so called M-transforms, for which polynomial equations are found. We exploit the methods of planar diagrammatics, enhanced to the non-Hermitian case, and free random variables, respectively; both are described in the appendices. As particular results of these two main equations, we find the singular behavior of the spectral densities near zero. Moreover, we propose a finite-size form of the spectral density of the product close to the border of its eigenvalues' domain. Also, led by the striking similarity between the two main equations, we put forward a conjecture about a simple relationship between the eigenvalues and singular values of any non-Hermitian random matrix whose spectrum exhibits rotational symmetry around zero.