Eigenvalues of one family of tridiagonal skew-self-adjoint Toeplitz matrices with complex perturbations on the corner

TL;DR

This paper analyzes the eigenvalues of a family of complex-perturbed skew-self-adjoint Toeplitz matrices, providing asymptotic formulas with high precision for large matrix sizes.

Contribution

It introduces an asymptotic formula for eigenvalues of perturbed Toeplitz matrices with complex corner perturbations, advancing spectral analysis methods.

Findings

Eigenvalues asymptotically distribute as 2i sin(x) for large n

Derived an asymptotic formula with O(1/n^3) residue for each eigenvalue

Eigenvalues follow a specific distribution pattern as n approaches infinity

Abstract

In this paper, we study the eigenvalues of the matrices $T_n(a)+\gamma E_{n,1,1}$ where $T_n(a)$ is the Toeplitz matrix with generating symbol $a(t)=t-t^{-1}$, $E_{n,1,1}$ is the $n\times n$ matrix whose upper left component is $1$ and the other components are zero, and $\gamma$ is a fixed complex number such that $0<|\gamma|<1$. As $n\to\infty$, the eigenvalues of these matrices are asymptotically distributed as the function $2 i \sin(x)$, $x\in[0,2\pi]$. Our main result is an asymptotic formula for every eigenvalue with a residue of the order $O(1/n^3)$.