Hermitian Matrix Diagonalization and its Symmetry Properties

A hermitian matrix can be parametrized by a set consisting of its determinant and the eigenvalues of its submatrices. We established a group of equations which connect these variables with the mixing parameters of diagonalization. These equations are simple in structure and manifestly invariant in form under the symmetry operations of dilatation, translation, rephasing and permutation. When applied to the problem of neutrino oscillation in matter they produced two new ``matter invariants" which are confirmed by available data.