On the Spectra of Certain Matrices Generated by Involutory Automorphisms

Abstract

Let A=P+K be an n × n complex matrix with $P = \frac12(A-HAH)$ and $K=\frac12(A+HAH)$, H being a unitary involution. Having characterized all unitary involutions, we investigate the spectral structure of P and K and, in particular, characterize the eigenvalues of K as zeros of a rational function, and prove that, for normal A, $\sigma(K)$ resides in the convex hull of $\sigma(A)$. We also demonstrate that this need not be true when A is not normal.