Relative Perturbation Theory for Quadratic Hermitian Eigenvalue Problem

TL;DR

This paper develops new relative perturbation bounds for eigenvalues and eigenvectors of regular quadratic Hermitian eigenvalue problems, with applications to mechanical models, supported by numerical experiments.

Contribution

It introduces novel relative perturbation bounds for quadratic Hermitian eigenvalue problems based on an equivalent Hermitian matrix pair formulation.

Findings

New bounds improve understanding of eigenvector and eigenvalue sensitivity.

Bounds are applicable to practical problems like mechanical systems with damping.

Numerical experiments demonstrate the bounds' effectiveness.

Abstract

In this paper, we derive new relative perturbation bounds for eigenvectors and eigenvalues for regular quadratic eigenvalue problems of the form $\lambda^2 M x + \lambda C x + K x = 0$, where $M$ and $K$ are nonsingular Hermitian matrices and $C$ is a general Hermitian matrix. We base our findings on new results for an equivalent regular Hermitian matrix pair $A-\lambda B$. The new bounds can be applied to many interesting quadratic eigenvalue problems appearing in applications, such as mechanical models with indefinite damping. The quality of our bounds is demonstrated by several numerical experiments.