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In this paper we bring to light an unprecedented property of the eigenvaluesof a matrix A with the eigenvalues and eigenvectors of a submatrix of A. Thisproperty can be used, through the technique developed here, to determine someof eigenvalues of A and, thus, reduce the degree of the characteristicpolynomial associated with this matrix. This reduction can occur in two ways:when restrictions between some elements of A are checked (or imposed) and/orwhen 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 thealgebraic calculation of eigenvalues and eigenvectors and how to share a set ofeigenvectors of a submatrix of A with matrix A itself, preserving norm,direction and sense.

Determining the eigenvalues of a square matrix through known information of its submatrix