  The current paper is a technical work that is focused on Lorentz violation
for Dirac fermions as well as neutrinos, described within the nonminimal
Standard-Model Extension. We intend to derive two theoretical results. The
first is the full propagator of the single-fermion Dirac theory modified by
Lorentz violation. The second is the dispersion equation for a theory of $N$
neutrino flavors that enables the description of both Dirac and Majorana
neutrinos. As the matrix structure of the neutrino field operator is very
involved for generic $N$, we will use sophisticated methods of linear algebra
to achieve our objectives. Our main finding is that the neutrino dispersion
equation has the same structure in terms of Lorentz-violating operators as that
of a modified single-fermion Dirac theory. The results will be valuable for
phenomenological studies of Lorentz-violating Dirac fermions and neutrinos.
