Effective Jordan-Chevalley decomposition (Q2900343)

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.