Symmetric texture-zero mass matrices with positive eigenvalues

Within the context of texture-zeros mechanism for fermionic mass matrices, we provide necessary and sufficientconditions on the characteristic polynomial coefficients such that it has real, simple and positive roots. We translatethese conditions in terms of invariants from congruent matrices. Then, all symmetric texture-zero matrices are countedand classified. Next we apply the result from the first part to analyse the three, two and one zero texture matrices in asystematic way. Finally we solve analytically the Vckm mixing matrix for the four zero sets; we also analyse the |Vckm| fora particular case of four zero, four zero-perturbed and three zero sets.