An application of Jordan canonical form to the proof of Cayley-Hamilton theorem

Abstract

The statement of Cayley-Hamilton theorem is that every square matrix satisfies its own characteristic equation. Cayley-Hamilton theorem holds both in a vector space over a field and in a module over a commutative ring. The general proof of Cayley-Hamilton theorem is based on the concepts of minimal polynomial and adjoint matrix of a linear map (for the details of the general proof, see Lang (2002), page 561, or Liesen and Mehrmann (2011), page 96, or Shurman). In the case of a diagonalizable matrix A over an algebraically closed field the proof becomes trivial because one can consider the diagonal form D of A and the relation for the k-th power matrix Ak = CDkC−1, where C is the matrix for the basis change to the basis of eigenvectors of A (for the details, see Sernesi (2000) or Lang (1987)). The aim of this paper is to extend the simple proof for diagonalizable matrices to the case of non-diagonalizable ones over a generic field. First, we obtain a proof for non-diagonalizable matrices over an algebraically closed field and then, by virtue of the properties of field extensions, we show that this proof also holds in the case of a generic field.