A new Gershgorin-type result for the localisation of the spectrum of matrices

TL;DR

This paper introduces a new, elementary Gershgorin-type theorem for matrix spectrum localization, outperforming previous methods and applicable to scalar and operator matrices with refined estimates.

Contribution

It presents a novel Gershgorin-type result using Schur complements, improving upon Cassini oval-based methods and applicable to both scalar and operator matrices.

Findings

Outperforms Cassini oval methods in spectrum localization

Provides estimates valid for scalar and operator matrices

Includes several refinements of existing results

Abstract

We present a Gershgorin's type result on the localisation of the spectrum of a matrix. Our method is elementary and relies upon the method of Schur complements, furthermore it outperforms the one based on the Cassini ovals of Ostrovski and Brauer. Furthermore, it yields estimates that hold without major differences in the cases of both scalar and operator matrices. Several refinements of known results are obtained.