Décomposition effective de Jordan-Chevalley et ses retombées en enseignement
| hal.structure.identifier | Institut de Mathématiques de Toulouse UMR5219 [IMT] | |
| dc.contributor.author | COUTY, Danielle | |
| hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
| dc.contributor.author | ESTERLE, Jean | |
| hal.structure.identifier | Mathématiques Fondamentales | |
| dc.contributor.author | ZAROUF, Rachid | |
| dc.date.accessioned | 2024-04-04T02:23:05Z | |
| dc.date.available | 2024-04-04T02:23:05Z | |
| dc.date.created | 2011-06-16 | |
| dc.date.issued | 2011-07-11 | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/189701 | |
| dc.description.abstractEn | The 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.iso | fr | |
| dc.publisher | Société Mathématique de France | |
| dc.title | Décomposition effective de Jordan-Chevalley et ses retombées en enseignement | |
| dc.type | Article de revue | |
| dc.subject.hal | Mathématiques [math]/Anneaux et algèbres [math.RA] | |
| dc.subject.hal | Mathématiques [math]/Mathématiques générales [math.GM] | |
| dc.subject.hal | Mathématiques [math]/Théorie des groupes [math.GR] | |
| dc.subject.hal | Mathématiques [math]/Analyse numérique [math.NA] | |
| dc.subject.hal | Mathématiques [math]/Théorie spectrale [math.SP] | |
| dc.identifier.arxiv | 1103.5020 | |
| bordeaux.journal | Gazette des Mathématiciens | |
| bordeaux.page | 29--49 | |
| bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
| bordeaux.issue | 129 | |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| bordeaux.peerReviewed | oui | |
| hal.identifier | hal-00525465 | |
| hal.version | 1 | |
| hal.popular | non | |
| hal.audience | Internationale | |
| dc.subject.it | Decomposition de Jordan | |
| dc.subject.it | Algorithme de Newton | |
| dc.subject.it | Decomposition de Chevalley | |
| dc.subject.it | Diagonalisable | |
| dc.subject.it | Nilpotent | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-00525465v1 | |
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