  It is well known that, for mass-action systems, complex-balanced equilibria
are asymptotically stable. For generalized mass-action systems, even if there
exists a unique complex-balanced equilibrium (in every stoichiometric class and
for all rate constants), it need not be stable.
  We first discuss several notions of matrix stability (on a linear subspace)
such as D-stability and diagonal stability, and then we apply our abstract
results to complex-balanced equilibria of generalized mass-action systems. In
particular, we show that linear stability (on the stoichiometric subspace and
for all rate constants) implies uniqueness. For cyclic networks, we
characterize linear stability (in terms of D-stability of the Jacobian matrix);
and for weakly reversible networks, we give necessary conditions for linear
stability (in terms of D-semistability of the Jacobian matrices of all cycles
in the network). Moreover, we show that, for classical mass-action systems,
complex-balanced equilibria are not just asymptotically stable, but even
diagonally stable (and hence linearly stable).
  Finally, we recall and extend characterizations of D-stability and diagonal
stability for matrices of dimension up to three, and we illustrate our results
by examples of irreversible cycles (of dimension up to three) and of reversible
chains and S-systems (of arbitrary dimension).
