A characterization of the Hurwitz stability of Metzler matrices

It is well known that a Hurwitz Metzler matrix is also diagonally stable. We obtain a necessary and sufficient condition for a matrix A to be diagonally stable from the Kalman-Yacubovich-Popov lemma. This condition is equivalent to requiring that a pair of LTI systems, of lower dimension, have a common Lyapunov function. This fact is made use of to derive very simple conditions for the Hurwitz stability of a Metzler matrix. These conditions are stated in terms of the signs of the diagonal entries of a sequence of lower dimensional matrices that are easily constructed.