Eigenvalues of tridiagonal Hermitian Toeplitz matrices with perturbations in the off-diagonal corners

TL;DR

This paper analyzes the eigenvalues of Hermitian Toeplitz matrices with specific off-diagonal perturbations, revealing how their distribution and extreme eigenvalues change depending on the perturbation magnitude.

Contribution

It provides asymptotic formulas for eigenvalues of perturbed Toeplitz matrices, especially highlighting the behavior of extreme eigenvalues for large matrix sizes.

Findings

Eigenvalues lie in [0,4] for |α| ≤ 1 and follow a specific distribution.

Extreme eigenvalues deviate from [0,4] when |α| > 1 and converge to limits based on α.

Derived asymptotic formulas facilitate numerical computation of eigenvalues.

Abstract

In this paper we study the eigenvalues of Hermitian Toeplitz matrices with the entries $2,-1,0,\ldots,0,-\alpha$ in the first column. Notice that the generating symbol depends on the order $n$ of the matrix. If $|\alpha|\le 1$, then the eigenvalues belong to $[0,4]$ and are asymptotically distributed as the function $g(x)=4\sin^2(x/2)$ on $[0,\pi]$. The situation changes drastically when $|\alpha|>1$ and $n$ tends to infinity. Then the two extreme eigenvalues (the minimal and the maximal one) lay out of $[0,4]$ and converge rapidly to certain limits determined by the value of $\alpha$, whilst all others belong to $[0,4]$ and are asymptotically distributed as $g$. In all cases, we transform the characteristic equation to a form convenient to solve by numerical methods, and derive asymptotic formulas for the eigenvalues.