Inertia of the stein transformation with respect to some nonderogatory matrices

Abstract

Let A be an n-by- n nonderogatory matrix all of whose eigenvalues lie on the unit circle, and let π and ν be nonnegative integers with π + ν = n. Let π′ and ν′ be positive integers and δ′ a nonnegative integer with π′ + ν′ + δ′ = n. In this paper we explore the existence of a Hermitian nonsingular matrix K with inertia (π, ν, 0), such that the Stein transformation of K corresponding to A, S A(K) = K − AKA∗ , is a Hermitian matrix with inertia (π′, ν′, δ′). The study is done by reducing A to Jordan canonical form. If C is an n-by- n nonderogatory matrix all of whose eigenvalues lie on the imaginary axis, then the results obtained for S A(K) are valid for the Lyapunov transformation, L C(K) = CK + KC∗ , of K corresponding to C.