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Décomposition effective de Jordan-Chevalley et ses retombées en enseignement

hal.structure.identifierInstitut de Mathématiques de Toulouse UMR5219 [IMT]
dc.contributor.authorCOUTY, Danielle
hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorESTERLE, Jean
hal.structure.identifierMathématiques Fondamentales
dc.contributor.authorZAROUF, Rachid
dc.date.accessioned2024-04-04T02:23:05Z
dc.date.available2024-04-04T02:23:05Z
dc.date.created2011-06-16
dc.date.issued2011-07-11
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/189701
dc.description.abstractEnThe purpose of this paper is to point the effectiveness of the Jordan-Chevalley decomposition, i.e. the decomposition of a square matrix $U$ with coefficients in a field $k$ containing the eigenvalues of $U$ as a sum $U=D+N,$ where $D$ is a diagonalizable matrix and $N$ a nilpotent matrix which commutes with $D.$ The most general version of this decomposition shows that every separable element $u$ of a $k$-algebra $A$ can be written in a unique way as a sum $u=d+n,$ where $d \in A$ is absolutely semi-simple and where $n\in A$ is nilpotent and commutes with $d.$ In fact an algorithm, due to C. Chevalley, allows to compute this decomposition: this algorithm is an adaptation to this context of the Newton method, which gives here the exact value of the absolutely semi-simple part $d$ of $u$ after a finite number of iterations. We illustrate the effectiveness of this method by computing the decomposition of a $15 \times 15$ matrix having eigenvalues of multiplicity 3 which are not computable exactly. We also discuss the other classical method, based on the chinese remainder theorem, which gives the Jordan-Chevalley decomposition under the form $u=q(u) +[u-q(u)],$ with $q(u)$ absolutely semi-simple, $u-q(u)$ nilpotent, where $q$ is any solution of a system of congruence equations related to the roots of a polynomial $p\in k[x]$ such that $p(u)=0.$ It is indeed possible to compute $q$ without knowing the roots of $p$ by applying the algorithm discussed above to $\pi(x),$ where $\pi: k[x] \to k[x]/pk[x]$ is the canonical surjection. We obtain this way after 2 iterations the polynomial $q$ of degree 14 associated to the $15\times 15$ matrix mentioned above. We justify by historical considerations the use of the name "Jordan-Chevalley decomposition", instead of the name "Dunford decomposition" which also appears in the literature, and we discuss multiplicative versions of this decomposition in semi-simple Lie groups. We conclude this paper showing why this decomposition should play a central role in a linear algebra course, even at a rather elementary level. Our arguments are based on a teaching experience of more than 10 years in an engineering school located on the Basque Coast.
dc.language.isofr
dc.publisherSociété Mathématique de France
dc.titleDécomposition effective de Jordan-Chevalley et ses retombées en enseignement
dc.typeArticle de revue
dc.subject.halMathématiques [math]/Anneaux et algèbres [math.RA]
dc.subject.halMathématiques [math]/Mathématiques générales [math.GM]
dc.subject.halMathématiques [math]/Théorie des groupes [math.GR]
dc.subject.halMathématiques [math]/Analyse numérique [math.NA]
dc.subject.halMathématiques [math]/Théorie spectrale [math.SP]
dc.identifier.arxiv1103.5020
bordeaux.journalGazette des Mathématiciens
bordeaux.page29--49
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.issue129
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
bordeaux.peerReviewedoui
hal.identifierhal-00525465
hal.version1
hal.popularnon
hal.audienceInternationale
dc.subject.itDecomposition de Jordan
dc.subject.itAlgorithme de Newton
dc.subject.itDecomposition de Chevalley
dc.subject.itDiagonalisable
dc.subject.itNilpotent
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-00525465v1
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