Local perturbations of block Toeplitz matrices

TL;DR

This paper investigates how local finite-rank perturbations affect the asymptotic spectral properties of block Toeplitz matrices, revealing continuous dependence of outliers and a new generalized Widom formula for characteristic polynomials.

Contribution

It introduces a generalized Widom formula for perturbed block Toeplitz matrices and analyzes the spectral impact of local finite-rank perturbations, extending existing spectral theory.

Findings

Continuous dependence of spectral outliers on local perturbations

Limit spectrum depends only on perturbation ranks

Extension of Widom's formula to perturbed block Toeplitz matrices

Abstract

This work is about the asymptotic spectral theory of tridiagonal Toeplitz matrices with matrix entries, with periodicity broken on a finite number of entries. Varying the ranks of these perturbations allow to interpolate between open boundary and circulant Toeplitz matrices. While the continuous part of the limit spectrum only depends on these ranks and no other aspect of the perturbation, the outliers of the spectrum depend continuously on the local perturbation. The proof is essentially based on a new generalized Widom formula for the characteristic polynomial. All this holds for Lebesgue almost all perturbed Toeplitz matrices, a fact that constitutes another important extension of Widom's work. The mathematical results are illustrated by numerics.