On the separation of eigenvalues by the permutation group

Abstract

Abstract Let A be an invertible 3 × 3 complex matrix. It is shown that there is a 3 × 3 permutation matrix P such that the product PA has at least two distinct eigenvalues. The nilpotent complex n × n matrices A for which the products PA with all symmetric matrices P have a single spectrum are determined. It is shown that for a n × n complex matrix there exists a permutation matrix P such that the product PA has at least two distinct eigenvalues.