On the diagonal stability for the Metzler matrices

We present a new proof of a classical theorem concerning Lyapunov functions for a stable Metzler matrix.The theorem says that the Lyapunov matrix can be taken diagonal.The proof is based on characterization of positive diagonal matrices via the Lyapunov inequality.This characterization reduces the search for the diagonal Lyapunov function to the search for a common Lyapunov function for an explicit set of stable matrices.