On the Lyapunov and Stein equations

Let L∈Cn×n and let H,K∈Cn×n 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.