Perturbation bounds of eigenvalues of Hermitian matrices with block structures

TL;DR

This paper develops new bounds on how much eigenvalues of Hermitian matrices with various block structures can change under perturbations, linking eigenvector components to eigenvalue sensitivity.

Contribution

It introduces novel perturbation bounds for Hermitian matrices with block structures, connecting eigenvector component sizes to eigenvalue stability.

Findings

Eigenvalues are less sensitive to perturbations when corresponding eigenvector components are small.

The bounds explain phenomena in Wilkinson's matrices and the effectiveness of early deflation in QR algorithms.

The approach applies to block 2x2, tridiagonal, and block tridiagonal Hermitian matrices.

Abstract

We derive new perturbation bounds for eigenvalues of Hermitian matrices with block structures. The structures we consider range from a standard 2-by-2 block form to block tridiagonal and tridigaonal forms. The main idea is the observation that an eigenvalue is insensitive to componentwise perturbations if the corresponding eigenvector components are small. We show that the same idea can be used to explain two well-known phenomena, one concerning extremal eigenvalues of Wilkinson's matrices and another concerning the efficiency of aggressive early deflation applied to the symmetric tridiagonal QR algorithm.