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 A, 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.

An Application of Jordan Canonical Form to the Proof of Cayley-Hamilton Theorem / De Marchis, R.; Grande, A.; Patri, S.; Saitta, D.. - In: ANNALI DEL DIPARTIMENTO DI METODI E MODELLI PER L'ECONOMIA, IL TERRITORIO E LA FINANZA ..... - ISSN 2385-0825. - (2019), pp. 43-48.

### 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 A, 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.
##### Scheda breve Scheda completa
2019
Jordan Canonical Form; Power of Jordan Matrix
01 Pubblicazione su rivista::01a Articolo in rivista
An Application of Jordan Canonical Form to the Proof of Cayley-Hamilton Theorem / De Marchis, R.; Grande, A.; Patri, S.; Saitta, D.. - In: ANNALI DEL DIPARTIMENTO DI METODI E MODELLI PER L'ECONOMIA, IL TERRITORIO E LA FINANZA ..... - ISSN 2385-0825. - (2019), pp. 43-48.
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11573/1355445`