Convergence of the spectral measure of non normal matrices

TL;DR

This paper demonstrates that adding a small Gaussian noise to large non-normal matrices causes their eigenvalue distribution to converge to the Brown measure of the limiting operator, revealing a regularization effect.

Contribution

It establishes a rigorous link between noise regularization and spectral measure convergence for large non-normal matrices, extending understanding of spectral stability.

Findings

Eigenvalue measures converge to the Brown measure after noise addition

Regularization by noise ensures spectral measure convergence

Results apply to matrices converging in *-moments to a regular element

Abstract

We discuss regularization by noise of the spectrum of large random non-Normal matrices. Under suitable conditions, we show that the regularization of a sequence of matrices that converges in *-moments to a regular element $a$, by the addition of a polynomially vanishing Gaussian Ginibre matrix, forces the empirical measure of eigenvalues to converge to the Brown measure of $a$.