On the problem of the vanishing discriminant

We show that the straightforward application of the discriminant to some physical problems may yield a trivial useless result. If the symmetry of the model matrix does not change with variations of the model parameter the discriminant may vanish for all values of the parameter due to degeneracy. We illustrate this problem by means of a simple $6\times 6$ matrix representation of an Hermitian Hamiltonian operator.