On the Lyapunov and Stein equations Original Research Article

Let image and let image be Hermitian matrices. The general inertia theorem gives a complete set of relations between the similarity class of L and the congruence class of H, when the Lyapunov equation LH + HL* = K is satisfied and K > 0. In this paper, we give some relations between the similarity class of L and the congruence class of K, when the Lyapunov equation is satisfied and H > 0. We also consider the corresponding problem with the Stein equation.