Gershgorin-Type Spectral Inclusions for Matrices

TL;DR

This paper develops new Gershgorin-type spectral inclusion sets for matrices, based on block-tridiagonal perturbations, providing sharper bounds for spectra and pseudospectra, especially for large Toeplitz matrices.

Contribution

It introduces a novel approach using block-tridiagonal perturbations for Gershgorin-type bounds, improving spectral estimates over previous methods.

Findings

Provides sharper spectral bounds for large Toeplitz matrices.

Extends Gershgorin-type inclusion sets to pseudospectra of submatrices.

Builds on recent work for bi-infinite matrices with new block-tridiagonal techniques.

Abstract

In this paper we derive sequences of Gershgorin-type inclusion sets for the spectra and pseudospectra of finite matrices. In common with previous generalisations of the classical Gershgorin bound for the spectrum, our inclusion sets are based on a block decomposition. In contrast to previous generalisations that treat the matrix as a perturbation of a block-diagonal submatrix, our arguments treat the matrix as a perturbation of a block-tridiagonal matrix, which can lead to sharp spectral bounds, as we show for the example of large Toeplitz matrices. Our inclusion sets, which take the form of unions of pseudospectra of square or rectangular submatrices, build on our own recent work on inclusion sets for bi-infinite matrices [Chandler-Wilde, Chonchaiya, Lindner, {\em J. Spectr. Theory} {\bf 14}, 719--804 (2024)].