Diagonalize the mass matrix

When we diagonalize the mass matrix, we use unitary bilinear transformation. Why unitary? Because we want no cross terms in the kinetic part, which can only be guaranteed by unitary transformation. Why bilinear? Because the left hand particles and the right hand particles are in principle independent, and we should not expect them to follow the same transformation. We know that the usual linear diagonalization is unique (if exists) from the secular equation. Now we can prove that the bilinear unitary diagonalization is unique in the terms of molds, but each eigenvalue can take an arbitrary phase. In the case of mass matrix, we require all the eigenvalues to be positive real numbers, so the diagonalization of the mass matrix is unique.