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

Recently, an upper bound for the number of distinct eigenvalues of a perturbed matrix was established in [{\sc P. Farrell}, {\em The number of distinct eigenvalues of a matrix after perturbation}, SIAM J. Matrix Anal. Appl., 37 (2016), pp. 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 give two refined upper bounds. Examples show the sharpness and effectiveness of our theoretical results.