The exact eigenstates of the neutrino mass matrix without CP-phase violation

In this paper we obtain the exact mass-eigenstates of the Majorana physical neutrinos. We start by taking into account a general $3\times3$ mass matrix without any CP-phase violation. It is then diagonalized by exactly solving an appropriate set of equations. The solution supplies straightforwardly the mass eigenvalues depending on the diagonal entries and mixing angles. Finally, the consequences of these analytical expressions are discussed assuming various phenomenological restrictions such as conserving the global lepton number $L=L_{e}-L_{\mu}-L_{\tau}$ and the $\mu-\tau$ interchange symmetry. The minimal absolute mass in the neutrino sector is also obtained since the two plausible scenarios invoked above are employed.