The number of distinct eigenvalues of a regular pencil and of a square matrix after rank perturbation

Abstract

We obtain lower and upper bounds for the number of distinct eigenvalues of a regular matrix pencil obtained from another regular pencil by a perturbation of arbitrary rank. Moreover, taking advantage of these bounds, we characterize the existence of a perturbation such that the perturbed pencil has prescribed number of distinct eigenvalues, over algebraically closed fields. We also prove that the results hold for the number of distinct eigenvalues of a square matrix obtained from another square matrix by a perturbation of arbitrary rank. Our bounds improve some other bounds existing in the literature.