Abstract
Let L ∈ C n × n and let H , K ∈ C n × 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.
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