An equilibrium problem for the limiting eigenvalue distribution of rational Toeplitz matrices

TL;DR

This paper investigates the asymptotic eigenvalue distribution of rational Toeplitz matrices, revealing that their limiting behavior can be described by a vector equilibrium problem, extending previous results for banded matrices.

Contribution

It introduces a new equilibrium framework for understanding the eigenvalue distribution of rational Toeplitz matrices, generalizing prior work on banded cases.

Findings

Weak limit of eigenvalue measure identified as a component of a vector equilibrium solution

Other components describe generalized eigenvalue behavior

Extends recent results for banded Toeplitz matrices

Abstract

We consider the asymptotic behavior of the eigenvalues of Toeplitz matrices with rational symbol as the size of the matrix goes to infinity. Our main result is that the weak limit of the normalized eigenvalue counting measure is a particular component of the unique solution to a vector equilibrium problem. Moreover, we show that the other components describe the limiting behavior of certain generalized eigenvalues. In this way, we generalize the recent results of Duits and Kuijlaars for banded Toeplitz matrices.