The authors consider a pair of positive linear time-invariant systems
are Metzler matrices. They study conditions for which both systems have a diagonal common quadratic Lyapunov function (CQLF). Their main result is that under the assumption that
are Hurwitz and have no zero entries the existence of a diagonal CQLF is equivalent to the condition that
is non-singular for all positiv definite diagonal matrices
. The paper is well structured and nicely written.