Gershgorin disks for multiple eigenvalues of non-negative matrices

TL;DR

This paper refines Gershgorin's circle theorem for non-negative matrices, showing that eigenvalues with higher geometric multiplicity are confined to smaller disks, using novel geometric inequalities.

Contribution

It introduces a new geometric rearrangement inequality and extends Gershgorin's theorem to eigenvalues with higher multiplicity in non-negative matrices.

Findings

Eigenvalues with geometric multiplicity ≥ 2 lie in smaller Gershgorin disks.

New geometric rearrangement inequalities for sums of higher-dimensional vectors.

Enhanced bounds for eigenvalue localization in non-negative matrices.

Abstract

Gershgorin's famous circle theorem states that all eigenvalues of a square matrix lie in disks (called Gershgorin disks) around the diagonal elements. Here we show that if the matrix entries are non-negative and an eigenvalue has geometric multiplicity at least two, then this eigenvalue lies in a smaller disk. The proof uses geometric rearrangement inequalities on sums of higher dimensional real vectors which is another new result of this paper.