The authors consider a pair of positive linear time-invariant systems

$\dot{x}={A}_{i}x$,

$i=1,2$, i.e.

${A}_{1},{A}_{2}$ 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

${A}_{1}$ and

${A}_{2}$ are Hurwitz and have no zero entries the existence of a diagonal CQLF is equivalent to the condition that

${A}_{1}+D{A}_{2}D$ is non-singular for all positiv definite diagonal matrices

$D$. The paper is well structured and nicely written.