  We revisit neutrino oscillations in constant matter density for a number of
different scenarios: three flavors with the standard Wolfenstein matter
potential, four flavors with standard matter potential and three flavors with
non-standard matter potentials. To calculate the oscillation probabilities for
these scenarios one must determine the eigenvalues and eigenvectors of the
Hamiltonians. We use a method for calculating the eigenvalues that is well
known, determination of the zeros of determinant of matrix $(\lambda I -H)$,
where H is the Hamiltonian, I the identity matrix and $\lambda$ is a scalar. To
calculate the associated eigenvectors we use a method that is little known in
the particle physics community, the calculation of the adjugate (transpose of
the cofactor matrix) of the same matrix, $(\lambda I -H)$. This method can be
applied to any Hamiltonian, but provides a very simple way to determine the
eigenvectors for neutrino oscillation in matter, independent of the complexity
of the matter potential. This method can be trivially automated using the
Faddeev-LeVerrier algorithm for numerical calculations. For the above scenarios
we derive a number of quantities that are invariant of the matter potential,
many are new such as the generalization of the Naumov-Harrison-Scott identity
for four or more flavors of neutrinos. We also show how these matter potential
independent quantities become matter potential dependent when off-diagonal
non-standard matter effects are included.
