Optimizing Gershgorin for Symmetric Matrices

TL;DR

This paper introduces an optimized Gershgorin-based method for bounding eigenvalues of symmetric matrices by efficiently choosing a shift parameter through linear programming, outperforming existing bounds in many cases.

Contribution

It develops a linear programming approach to optimize Gershgorin bounds for symmetric matrices, providing improved eigenvalue estimates and extending to nonlinear estimators under symmetry conditions.

Findings

Efficient linear program for optimal Gershgorin shift.

Outperforms existing eigenvalue bounds for large classes of matrices.

Provides a new eigenvalue bound based on graph degrees.

Abstract

The Gershgorin Circle Theorem is a well-known and efficient method for bounding the eigenvalues of a matrix in terms of its entries. If $A$ is a symmetric matrix, by writing $A = B + x{\bf 1}$, where ${\bf 1}$ is the matrix with unit entries, we consider the problem of choosing $x$ to give the optimal Gershgorin bound on the eigenvalues of $B$, which then leads to one-sided bounds on the eigenvalues of $A$. We show that this $x$ can be found by an efficient linear program (whose solution can in may cases be written in closed form), and we show that for large classes of matrices, this shifting method beats all existing piecewise linear or quadratic bounds on the eigenvalues. We also apply this shifting paradigm to some nonlinear estimators and show that under certain symmetries this also gives rise to a tractable linear program. As an application, we give a novel bound on the lowest eigenvalue of a adjacency matrix in terms of the "top two" or "bottom two" degrees of the corresponding graph, and study the efficacy of this method in obtaining sharp eigenvalue estimates for certain classes of matrices.