  A fundamental problem in the theory of PT-invariant quantum systems is to
determine whether a given system `respects' this symmetry or not. If not, the
system usually develops non-real eigenvalues. It is shown in this contribution
how to algorithmically detect the existence of complex eigenvalues for a given
PT-symmetric matrix. The procedure uses classical results from stability theory
which qualitatively locate the zeros of real polynomials in the complex plane.
The interest and value of the present approach lies in the fact that it avoids
diagonalization of the Hamiltonian at hand.
