The Rotation of Eigenspaces of Perturbed Matrix Pairs

TL;DR

This paper develops new estimates for how invariant subspaces of positive definite matrix pairs rotate under perturbations, especially in parameter-dependent eigenvalue problems, providing sharper bounds than previous methods.

Contribution

It introduces a novel approach to relative perturbation theory that yields sharper estimates for spectral subspace rotation in parameter-dependent matrix pairs.

Findings

New estimates for spectral subspace rotation are derived.

Estimates are sharp as functions of the parameter.

The approach improves upon existing perturbation bounds.

Abstract

We revisit the relative perturbation theory for invariant subspaces of positive definite matrix pairs. As a prototype model problem for our results we consider parameter dependent families of eigenvalue problems. We show that new estimates are a natural way to obtain sharp --- as functions of the parameter indexing the family of matrix pairs --- estimates for the rotation of spectral subspaces.