  In order to obtain stable and accurate general relativistic simulations,
re-formulations of the Einstein equations are necessary. In a series of our
works, we have proposed using eigenvalue analysis of constraint propagation
equations for evaluating violation behavior of constraints. In this article, we
classify asymptotical behaviors of constraint-violation into three types
(asymptotically constrained, asymptotically bounded, and diverge), and give
their necessary and sufficient conditions. We find that degeneracy of
eigenvalues sometimes leads constraint evolution to diverge (even if its
real-part is not positive), and conclude that it is quite useful to check the
diagonalizability of constraint propagation matrices. The discussion is general
and can be applied to any numerical treatments of constrained dynamics.
