Abstract

A perturbation result, due to Rado, shows how to modify r eigenvalues of a matrix of order n, via a perturbation of rank r ≤ n, without changing any of the n − r remaining eigenvalues. This result extended a previous one, due to Brauer, on perturbations of rank 1. Both results have been exploited in connection with the nonnegative inverse eigenvalue problem and the nonnegative inverse elementary divisors problem. In this paper, we use the Rado result from a more general point of view, constructing a family of matrices with prescribed spectrum and elementary divisors, generalizing previous results. We also apply our results to the nonnegative pole assigment problem.