Jordan-Chevalley decomposition

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In mathematics, the Jordan–Chevalley decomposition expresses a linear operator as the sum of its semisimple part and its nilpotent parts.

Consider linear operators on a finite-dimensional vector space. An operator is semisimple if the roots of its minimal polynomial are all distinct (if the underlying field is algebraically closed, this is the same as the requirement that the operator be diagonalizable). An operator $~x~$ is nilpotent if some power $~x^m~$ of it is the zero operator.

Now, let $~x~$ be any operator. The Jordan–Chevalley decomposition expresses $~x~$ as a sum

$~x = x_{\mathrm{ss}} + x_{\mathrm{n}}~$ ,

where $~x_{\mathrm{ss}}~$ is semisimple, $~x_{\mathrm{n}}~$ is nilpotent, and $~x_{\mathrm{ss}}~$ and $~x_{\mathrm{n}}~$ are polynomials in $~x~$ (in particular they commute). This decomposition is unique.