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Localization of complementarity eigenvalues

Antonio Sasaki, Sophie Demassey, Valentina Sessa

TL;DR

This work extends Gershgorin-type localization for the symmetric EiCP from the identity matrix case to general $B$ that is positive definite and strictly diagonally dominant. It derives two localization sets, $K_1$ (one-row) and $K_2$ (two-row, under copositivity of $A$), with $K_2$ strictly tighter than $K_1$, and provides computable bounds for the extreme complementarity eigenvalues from these sets. The authors compare these enclosures with the classical generalized spectrum, showing that EiCP eigenvalues lie inside the generalized spectrum but that there is no universal dominance between the bounds; in some scenarios $K_2$ can be significantly sharper. The results offer fast certificates for candidate EiCP solutions and open directions for multi-row localization and tensor EiCP extensions, with potential applications in copositive analysis and stability studies.

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.

Localization of complementarity eigenvalues

TL;DR

This work extends Gershgorin-type localization for the symmetric EiCP from the identity matrix case to general that is positive definite and strictly diagonally dominant. It derives two localization sets, (one-row) and (two-row, under copositivity of ), with strictly tighter than , and provides computable bounds for the extreme complementarity eigenvalues from these sets. The authors compare these enclosures with the classical generalized spectrum, showing that EiCP eigenvalues lie inside the generalized spectrum but that there is no universal dominance between the bounds; in some scenarios can be significantly sharper. The results offer fast certificates for candidate EiCP solutions and open directions for multi-row localization and tensor EiCP extensions, with potential applications in copositive analysis and stability studies.

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.
Paper Structure (7 sections, 15 theorems, 52 equations)

This paper contains 7 sections, 15 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Localization sets for EiCP solutions
  3. Comparing the localization sets
  4. Lower bounds and upper bounds
  5. Comparison with the generalized spectrum
  6. Discussion
  7. Conclusions

Key Result

Theorem 2.2

Let $\mathbf{A},\mathbf{B}\in\mathbb{S}_n$, and assume that $\mathbf{B}$ is positive definite and strictly diagonally dominant. Let $\lambda \in \mathbb R$ be a complementarity eigenvalue of $(\mathbf{A},\mathbf{B})$, then

Theorems & Definitions (36)

  • Definition 1.1: Diagonal dominance
  • Definition 1.2: Positivity
  • Definition 1.3: Copositivity
  • Definition 2.1
  • Theorem 2.2: One-row localization
  • Lemma 2.3
  • proof
  • proof : Proof of Theorem \ref{['thm:one-row-theorem']}.
  • Corollary 2.4
  • proof
  • ...and 26 more