Common Lyapunov solutions for two matrices whose difference has rank one

Real stable matrices A and B with rank of A-B equal to one have a common Lyapunov solution if and only if their product AB has no real negative eigenvalue. This was proved by Shorten and Narendra [R.N. Shorten, K.S. Narendra, On common quadratic Lyapunov functions for pairs of stable LTI systems whose system matrices are in companion form, IEEE Trans. Automat. Control 48 (4) (2003) 618–621], whose proof is based on the fundamental results of Kalman on Luré’s problem. In this paper we give an alternative proof of this result and its generalization to the general regular inertia case, and to the case when the matrices A and B are complex.