On the separation of eigenvalues by the permutation group

Abstract

Let A be an invertible × complex matrix. It is shown that there is a × 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 A = [a ij ] n i,j= with | i,j a ij | ≤ n n | det A| there exists a permutation matrix P such that the product PA has at least two distinct eigenvalues.