  We consider the class of conservative-dissipative ODE systems, which is a
subclass of Lyapunov stable, linear time-invariant ODE systems. We characterize
asymptotically stable, conservative-dissipative ODE systems via the
hypocoercivity (theory) of their system matrices. Our main result is a concise
characterization of the hypocoercivity index (an algebraic structural property
of matrices with positive semi-definite Hermitian part introduced in
Achleitner, Arnold, and Carlen (2018)) in terms of the short time behavior of
the propagator norm for the associated conservative-dissipative ODE system.
