Some Applications of the Hamilton-Cayley Theorem

Abstract

We saw in a previous chapter that the characteristic polynomial of a matrix and its zeros, the eigenvalues of the matrix, can give precious information. The purpose of this chapter is to present a powerful theorem due to Hamilton and Cayley that gives an even stronger relation between a matrix and its characteristic polynomial: in a certain sense, the matrix is a “root” of its characteristic polynomial. After presenting a proof of this theorem, we investigate some interesting applications.