D\'ecomposition effective de Jordan-Chevalley et ses retomb\'ees en enseignement

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.