  Given a positive definite symmetric matrix in one of the groups SL(n, R) or
Sp(n, R), we analyse its actions on Flag Manifolds, proving these are
diffeomorphisms which admit as strict Lyapunov functions a special class of
quadratic functions, all of them Z2 -perfect Morse functions. Stratifications
on the Flag Manifolds are provided which are invariant under both the
diffeomorphisms and the gradient flows of the Lyapunov functions.
