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

R. De Marchis;A. Grande;S. Patri;D. Saitta
2019

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.
File allegati a questo prodotto
File Dimensione Formato  
DeMarchis_Application-Jordan_2019.pdf

accesso aperto

Tipologia: Versione editoriale (versione pubblicata con il layout dell'editore)
Licenza: Creative commons
Dimensione 409.66 kB
Formato Adobe PDF
409.66 kB Adobe PDF Visualizza/Apri PDF

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11573/1355445
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? ND
social impact