Interior eigenvalue density of Jordan matrices with random perturbations

TL;DR

This paper analyzes how the eigenvalue distribution of large Jordan matrices is affected by small random Gaussian perturbations, providing a detailed asymptotic description of the eigenvalue density inside the main circle.

Contribution

It offers a precise asymptotic characterization of the expected eigenvalue density inside the circle for large Jordan matrices with random perturbations, extending previous probabilistic results.

Findings

Eigenvalues concentrate near a circle as matrix size grows

Expected eigenvalue density inside the circle is asymptotically described

Most eigenvalues are close to the circle with high probability

Abstract

We study the eigenvalue distribution of a large Jordan block subject to a small random Gaussian perturbation. A result by E.B. Davies and M. Hager shows that as the dimension of the matrix gets large, with probability close to $1$, most of the eigenvalues are close to a circle. We study the expected eigenvalue density of the perturbed Jordan block in the interior of that circle and give a precise asymptotic description.