An improved upper bound for the number of distinct eigenvalues of a matrix after perturbation

TL;DR

This paper improves the upper bound on the number of distinct eigenvalues of a matrix after perturbation, which is crucial for estimating Krylov iteration counts in solving perturbed linear systems.

Contribution

The paper introduces a tighter upper bound for the number of distinct eigenvalues post-perturbation, enhancing previous estimates and their applications.

Findings

New upper bound for eigenvalues after perturbation

Applications to Krylov subspace methods

Enhanced estimates for linear system solutions

Abstract

An upper bound for the number of distinct eigenvalues of a perturbed matrix has been recently established by P. E. Farrell [1, Theorem 1.3]. The estimate is the central result in Farrell's work and can be applied to estimate the number of Krylov iterations required for solving a perturbed linear system. In this paper, we present an improved upper bound for the number of distinct eigenvalues of a matrix after perturbation. Furthermore, some results based on the improved estimate are presented.