2D Eigenvalue Problem I: Existence and Number of Solutions

TL;DR

This paper introduces a two-dimensional eigenvalue problem (2DEVP) for Hermitian matrix pairs, exploring its fundamental properties, conditions for finite solutions, and connections to eigenvalue optimization and stability analysis.

Contribution

It provides the first comprehensive analysis of 2DEVP, including existence, solution count, and variational characterizations, linking it to eigenvalue optimization and stability problems.

Findings

Established conditions for the existence of 2D-eigenvalues.

Derived variational characterizations of 2DEVP solutions.

Connected 2DEVP properties to eigenvalue optimization and stability measures.

Abstract

A two dimensional eigenvalue problem (2DEVP) of a Hermitian matrix pair $(A, C)$ is introduced in this paper. The 2DEVP can be viewed as a linear algebraic formulation of the well-known eigenvalue optimization problem of the parameter matrix $H(\mu) = A - \mu C$. We present fundamental properties of the 2DEVP such as the existence, the necessary and sufficient condition for the finite number of 2D-eigenvalues and variational characterizations. We use eigenvalue optimization problems from the minmax of two Rayleigh quotients and the computation of distance to instability to show their connections with the 2DEVP and new insights of these problems derived from the properties of the 2DEVP.