  Neutrinos stand out among elementary particles through their unusually small
masses. Various seesaw mechanisms attempt to explain this fact. In this work
applying insights from matrix theory we are in a position to treat variants of
seesaw mechanisms in a general manner. Specifically, using Weyl's inequalities
we discuss and rigorously prove under which conditions the seesaw framework
leads to a mass spectrum with exactly three light neutrinos. We find an
estimate on the mass of heavy neutrinos to be the mass obtained by neglecting
light neutrinos shifted at most by the maximal strength of the coupling to the
light neutrino sector. We provide analytical conditions allowing to prescribe
that precisely two out of five neutrinos are heavy. For higher-dimensional
cases the inverse eigenvalue methods are used. In particular, for the CP
invariant scenarios we show that if the neutrino sector has a valid mass matrix
after neglecting the light ones, i.e. the respective mass submatrix is positive
definite, then large masses are provided by matrices with large elements
accumulated on the diagonal. Finally, the Davis-Kahan theorem is used to show
how masses affect the rotation of light neutrino eigenvectors from the standard
Euclidean basis. This general observation concerning neutrino mixing together
with results on the mass spectrum properties opens directions for further
neutrino physics studies using matrix analysis.
