# Random Perturbations of Matrix Polynomials

## TL;DR

This paper studies how the eigenvalues of large random matrix polynomials are affected by low-rank perturbations, deriving formulas for their limiting behavior and illustrating with examples and simulations.

## Contribution

It provides a new formula for the resolvent limit of sum of random and low-rank matrix polynomials, extending understanding of eigenvalue localization.

## Key findings

- Eigenvalues are localized using the derived resolvent formula.
- Theoretical results are validated through numerical simulations.
- Three specific instances demonstrate the applicability of the results.

## Abstract

A sum of a large-dimensional random matrix polynomial and a fixed low-rank matrix polynomial is considered. The main assumption is that the resolvent of the random polynomial converges to some deterministic limit. A formula for the limit of the resolvent of the sum is derived and the eigenvalues are localised. Three instances are considered: a low-rank matrix perturbed by the Wigner matrix, a product $HX$ of a fixed diagonal matrix $H$ and the Wigner matrix $X$ and a special matrix polynomial. The results are illustrated with various examples and numerical simulations.