  A linear dynamical system is called $k$-positive if its dynamics maps the set
of vectors with up to $k-1$ sign variations to itself. For $k=1$, this reduces
to the important class of positive linear systems. Since stable positive linear
time-invariant (LTI) systems always admit a diagonal quadratic Lyapunov
function, i.e. they are diagonally stable, we may expect that this holds also
for stable $k$-positive systems. We show that, in general, this is not the case
both in the continuous-time (CT) and discrete-time (DT) case. We then focus on
DT $k$-positive linear systems and introduce the new notion of DT $k$-diagonal
stability. It is shown that this is a necessary condition for standard DT
diagonal stability. We demonstrate an application of this new notion to the
analysis of a class of DT nonlinear systems.
