Inertia of the stein transformation with respect to some nonderogatory matrices Original Research Article

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, SA(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 SA(K) are valid for the Lyapunov transformation, LC(K) = CK + KC*, of K corresponding to C.