Localization of complementarity eigenvalues

TL;DR

This paper develops new localization sets for complementarity eigenvalues of symmetric matrix pairs, extending existing methods to cases where B is not the identity and comparing bounds with classical eigenvalues.

Contribution

It introduces two new localization sets for complementarity eigenvalues, extending previous results to broader matrix conditions, especially when B is not the identity.

Findings

Derived two localization sets for complementarity eigenvalues.

Extended He-Liu-Shen sets to cases with non-identity B.

Compared bounds from new sets with classical eigenvalues.

Abstract

Let A, B be symmetric n x n real matrices with B positive definite and strictly diagonally dominant. We derive two localization sets for the complementarity eigenvalues of (A, B), the tightest one assuming additionally that A is copositive. This extends He-Liu-Shen sets to the case where B is not the identity. Moreover, we compare the computable bounds obtained from these new sets with the extreme classical generalized eigenvalues.