The eigenvalues and eigenvectors of finite-rank normal perturbations of large rotationally invariant non-Hermitian matrices

TL;DR

This paper analyzes how finite-rank normal perturbations affect the eigenvalues and eigenvectors of large rotationally invariant non-Hermitian matrices, extending classical BBP results to a broader non-Hermitian context.

Contribution

It generalizes the BBP framework to non-Hermitian matrices, characterizing outlier eigenvalues and eigenvector behavior under finite-rank normal perturbations.

Findings

Outlier eigenvalues emerge and fluctuate according to the perturbation.

Eigenvector behavior is characterized in the presence of finite-rank perturbations.

The framework unifies Hermitian and non-Hermitian cases.

Abstract

We study finite-rank normal deformations of rotationally invariant non-Hermitian random matrices. Extending the classical Baik-Ben Arous-P\'ech\'e (BBP) framework, we characterize the emergence and fluctuations of outlier eigenvalues in models of the form $\mathbf{A} + \mathbf{T}$, where $\mathbf{A}$ is a large rotationally invariant non-Hermitian random matrix and $\mathbf{T}$ is a finite-rank normal perturbation. We also describe the corresponding eigenvector behavior. Our results provide a unified framework encompassing both Hermitian and non-Hermitian settings, thereby generalizing several known cases.