  We consider $k$-positive linear systems, that is, systems that map the set of
vectors with up to $k-1$ sign variations to itself. For $k=1$, this reduces to
positive linear systems. It is well-known that stable positive linear time
invariant (LTI) systems admit a diagonal Lyapunov function. This property has
many important implications. A natural question is whether stable $k$-positive
systemsalso admit a diagonal Lyapunov function. This paper shows that, in
general, the answer is no. However, for both continuous-time and discrete-time
$n$-dimensional systems that are $(n-1)$-positive we provide a sufficient
condition for diagonal stability.
