Refined bounds on the number of distinct eigenvalues of a matrix after low-rank update

Abstract

ABSTRACT The eigenproblems of low-rank updated matrices are of crucial importance in many applications. Recently, an upper bound on the number of distinct eigenvalues of a low-rank updated matrix was established in [Farrell P. SIAM J Matrix Anal Appl. 2016:37;572–576]. 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 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.