Autori: Patrì, Stefano, De Marchis, Roberto, Grande, Antonio, Saitta, Daniela
Titolo: An application of Jordan canonical form to the proof of Cayley-hamilton theorem
Periodico: Annali del Dipartimento di metodi e modelli per l'economia, il territorio e la finanza
Anno: 2019 - Volume: 20 - Pagina iniziale: 43 - Pagina finale: 48

The statement of Cayley-Hamilton theorem is the 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 be comes 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 ofnon-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




SICI: 2385-0825(2019)20<43:AAOJCF>2.0.ZU;2-T
Testo completo: https://web.uniroma1.it/memotef/sites/default/files/Annali-2019_43-48_DeMarchisR_GrandeA_Patr%C3%ACS_SaittaD.pdf

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