  Techniques are developed for constructing amplitudes of neutrino-related
processes in terms of the neutrino mass matrix, with no reference to the
neutrino mixing matrix. The amplitudes of neutrino oscillations in vacuum and
medium, quasi-elastic neutrino scattering, $\beta$ decays and double-$\beta$
decays are considered. The proposed approach makes extensive use of Frobenius
covariants within the framework of Sylvester's theorem on matrix functions. The
in-medium dispersion laws are found in terms of elementary functions for three
flavors of Majorana neutrinos as an application of the developed formalism. The
in-medium dispersion laws for Dirac neutrinos can be determined in the general
case by searching for the roots of a polynomial of degree 6. In the rest frame
of baryon matter, the minimum energy of both Majorana and Dirac neutrinos is
achieved at a finite neutrino momentum. In such cases, Dirac neutrinos occupy a
hollow Fermi sphere at zero temperature and low densities. Fitting experimental
data in terms of the neutrino mass matrix can provide better statistical
accuracy in determining the neutrino mass matrix compared to methods using the
neutrino mixing matrix at intermediate stages.
