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

Abstract

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.