Effective Jordan-Chevalley decomposition
type: Item
instance of: scholarly article
zbMATH Open document ID: 1251.15018
author: Danielle Couty
author: Jean Esterle
author: Rachid Zarouf
publication date: 2012-07-23T00:00:00Z
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 65F05
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 6059369
zbMATH Keywords: Jordan-Chevalley decompositon
zbMATH Keywords: Jordan canonical form
zbMATH Keywords: algorithm
zbMATH Keywords: computer algebra package Maple
zbMATH Keywords: powers and exponentials of matrices
zbMATH Keywords: semisimple Lie groups
describes a project that uses: Maple
MaRDI profile type: Publication
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title: Effective Jordan-Chevalley decomposition
published in: Gazette des Mathématiciens
review text: Let \(U\) be a square matrix over a field \(k\), and suppose that its minimal polynomial \(p\) can be factored \(p_{1}^{m_{1}}\dots p_{r}^{m_{r}}\) where the \(p_{i}\) are distinct monic irreducible polynomials over \(k\) and \(m_{i}\geq1\). Put \(\tilde{p}:=p_{1}\dots p_{r}\) and assume that \(\tilde{p}\) is separable (so \(\tilde{p}=p/GCD(p,p^{\prime})\)). Then it is well known that \(U\) can be written uniquely in the form \(D+N\) where \(D\) is diagonalizable over the algebraic closure of \(k\), \(N\) is nilpotent and \(DN=ND\) (the authors call this the Jordan-Chevalley decomposition rather than the Jordan decomposition because of its existence in more general situations which they consider). NEWLINENEWLINE\textit{C. Chevalley} [Théorie des groupes de Lie. Tome II. Groupes algébriques. Actualités scientifiques et industrielles. 1152. Publ. Inst. Math. Univ. Nancago, I. Paris: Hermann \& Cie. (1951; Zbl 0054.01303)] has shown that we can compute this decomposition over \(k\) without factoring or computing any eigenvalues as follows. Choose the polynomial \(q\) so that \(q\tilde{p}^{\prime}\equiv 1~(\text{mod}\) \(p/\tilde{p})\). Consider the Newtonian recurrence \(D_{0}:=U\) and \(D_{n+1}:=D_{n}-\tilde{p}(D_{n})q(D_{n})\) (\(n=0,1,\dots\)). If \(n_{0}\) is the least integer such that \(2^{n_{0}}\geq m_{i}\) for all \(i\), then \(\tilde{p}(D_{n_{0}})=0\) and \(D:=D_{n_{0}}\), \(N:=U-D\) gives the Jordan-Chevalley decomposition of \(U\).NEWLINENEWLINEThe present paper is expository. It illustrates how Chevalley's algorithm can be used to compute the decomposition with the computer algebra package Maple and how it can be applied to compute powers and exponentials of matrices. The paper also provides an interesting account of the history of the decomposition in more general situations, in particular in semisimple Lie groups.
reviewed by: John D. Dixon
description: scientific article; zbMATH DE number 6059369