Refined Bounds on the Number of Distinct Eigenvalues of a Matrix After Perturbation
Provides tighter theoretical guarantees for eigenvalue behavior under low-rank perturbations, which can impact the efficiency of Krylov subspace methods for practitioners solving perturbed linear systems.
The paper refines upper bounds on the number of distinct eigenvalues of a matrix after a low-rank perturbation, improving upon existing results. The new bounds rely only on the original matrix and the update, and examples demonstrate their superiority.
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.