Strict positive realness and the existence of diagonal Lyapunov functions

The problem of determining whether or not a stable LTI dynamical system has a Lyapunov function V(x) = x^TPx, with a diagonal matrix P > 0, referred to as a diagonal Lyapunov function, has attracted a great deal of attention in the past. The results obtained thus far are, for the most part, algebraic in nature. In this paper, using a geometric approach, and the concept of common Lyapunov functions, necessary and sufficient conditions for the existence of a diagonal Lyapunov function are given in terms of strict positive realness of a transfer function matrix