On Some Bounds on the Perturbation of Invariant Subspaces of Normal Matrices with Application to a Graph Connection Problem

TL;DR

This paper derives tighter bounds on how much invariant subspaces of normal matrices can change under perturbations, using spectral information, and applies these bounds to a graph perturbation problem.

Contribution

It introduces improved bounds on invariant subspace perturbations for normal matrices, surpassing the Davis-Khan $ ext{sin} heta$ theorem, with an application to graph problems.

Findings

Tighter bounds on invariant subspace perturbation for normal matrices.

Bounds depend on spectra of unperturbed and perturbed matrices.

Application demonstrated in a graph connection problem.

Abstract

We provide upper bounds on the perturbation of invariant subspaces of normal matrices measured using a metric on the space of vector subspaces of $\mathbb{C}^n$ in terms of the spectrum of both the unperturbed \& perturbed matrices, as well as, spectrum of the unperturbed matrix only. The results presented give tighter bounds than the Davis-Khan $\sin\Theta$ theorem. We apply the result to a graph perturbation problem.