On the Lyapunov and Stein Equations, II

Let L∈Cn×n and let H,K∈Cn×n be Hermitian matrices.Some already known results, including the general inertia theorem, give partial answers to the following problem: find a complete set of relations between the similarity class of L and the congruence classes of H and K, when the Lyapunov equation LH+HL*=K is satisfied.In this paper, we solve this problem when L is nonderogatory, H is nonsingular and K has at least one eigenvalue with positive real part and one eigenvalue with negative real part. Our result generalizes a previous paper by L. M. DeAlba.The corresponding problem with the Stein equation follows easily using a Cayley transform.