On the existence of a common quadratic Lyapunov function for a rank one difference

Suppose that A and B are real Hurwitz matrices, and that their difference A − B is rank one. Then A and B have a common quadratic Lyapunov function if and only if the product AB has no real negative eigenvalue. This result is due to Shorten and Narendra, who showed that it follows as a consequence of the Kalman–Yacubovich–Popov lemma and the solution of the Lur’e problem. Here we present a new and independent proof based on results from convex analysis and the theory of moments.