Refined Bounds on the Number of Distinct Eigenvalues of a Matrix After Perturbation

TL;DR

This paper refines bounds on the number of distinct eigenvalues of a matrix after low-rank perturbation, improving theoretical estimates relevant for Krylov subspace methods and analyzing singular value changes.

Contribution

The paper introduces refined bounds on the number of distinct eigenvalues after perturbation, relying solely on matrix and update information, with demonstrated improvements over previous bounds.

Findings

Refined bounds outperform existing estimates.

Bounds depend only on matrix and low-rank update information.

Analysis of singular values after perturbation included.

Abstract

The eigenproblem of low-rank updated matrices are of crucial importance in many applications. Recently, an upper bound on the number of distinct eigenvalues of a perturbed matrix was established. The result can be applied to estimate the number of Krylov iterations required for solving a perturbed linear system. In this paper, we revisit this problem and establish some refined bounds. Some {\it a prior} upper bounds that only rely on the information of the matrix in question and the low-rank update are provided. Examples show the superiority of our theoretical results over the existing ones. The number of distinct singular values of a matrix after perturbation is also investigated.