Quadratic perturbation bounds for generalized eigenvalue problems

TL;DR

This paper establishes quadratic bounds for eigenvalue perturbations in generalized Hermitian and non-Hermitian problems, improving understanding of eigenvalue stability under perturbations.

Contribution

It introduces quadratic perturbation bounds for generalized eigenvalue problems and refines the first-order expansion for multiple eigenvalues.

Findings

Quadratic bounds are proportional to the square of perturbation norms.

Derived tighter first-order perturbation expansion for multiple eigenvalues.

Presented quadratic bounds applicable to non-Hermitian cases.

Abstract

We prove quadratic eigenvalue perturbation bounds for generalized Hermitian eigenvalue problems. The bounds are proportional to the square of the norm of the perturbation matrices divided by the gap between the spectrums. Using the results we provide a simple derivation of the first-order perturbation expansion of a multiple eigenvalue, whose trailing term is tighter than known results. We also present quadratic bounds for the non-Hermitian case.