  In this paper we bring to light an unprecedented property of the eigenvalues
of a matrix A with the eigenvalues and eigenvectors of a submatrix of A. This
property can be used, through the technique developed here, to determine some
of eigenvalues of A and, thus, reduce the degree of the characteristic
polynomial associated with this matrix. This reduction can occur in two ways:
when restrictions between some elements of A are checked (or imposed) and/or
when some eigenvalues of the submatrix of A are degenerate. As an application,
we show how to obtain matrices of arbitrary size, not trivial, that allow the
algebraic calculation of eigenvalues and eigenvectors and how to share a set of
eigenvectors of a submatrix of A with matrix A itself, preserving norm,
direction and sense.
