On the perturbations of well separated matrices

TL;DR

This paper studies how eigenvalues of well separated matrices are affected by perturbations, providing bounds and insights into eigenvector behavior, with applications to Perron vector estimation.

Contribution

It introduces a quadratic oval region for eigenvalue errors under perturbations of well separated matrices and offers bounds based on Gershgorin circle parameters.

Findings

Eigenvalue perturbation region is a quadratic oval for well separated matrices.

Interlacing theorem for eigenvalues when separation is O(n).

Eigenvector condition number bounds relate to matrix separation and aid in Perron vector estimation.

Abstract

A matrix is well separated if all its Gershgorin circles are away from the unit circle and they are separated from each other. In this article, the region of relative errors in the eigenvalues is obtained as a quadratic oval for non diagonal perturbation of well seperated matrices. Thus giving a computable relative error bound in terms of Gershgorin circle parameters. When the separation is $O(n)$ and the matrix is positive definite, an interlacing theorem for the eigenvalues under perturbation is presented. Further when the separation is $O(n^2 )$, condition number of the eigenvector matrix is upper bounded to obtain the region of perturbed eigenvalue. Numerical results show the relation between diagonal entries and the magnitude of the eigenvector entries even when the matrix is not so well separated. We exploit this trend in estimating the Perron vector using power method.