Eigenvalue spectrum support of paired random matrices with pseudo-inverse

TL;DR

This paper investigates the eigenvalue spectrum support of the product of a Gaussian rectangular matrix and the pseudo-inverse of another, revealing insights into their spectral properties in random matrix theory.

Contribution

It provides a novel analysis of the eigenvalue spectrum support for paired Gaussian matrices involving pseudo-inverses, expanding understanding in random matrix theory.

Findings

Characterization of the eigenvalue spectrum support for XY†

Insights into spectral behavior of Gaussian paired matrices

Extension of spectral analysis to pseudo-inverse related products

Abstract

The Moore-Penrose pseudo-inverse $X^\dagger$, defined for rectangular matrices, naturally emerges in many areas of mathematics and science. For a pair of rectangular matrices $X, Y$ where the corresponding entries are jointly Gaussian and i.i.d., we analyse the support of the eigenvalue spectrum of $XY^\dagger$.