Jordan–Chevalley decomposition

In mathematics, specifically linear algebra, the Jordan–Chevalley decomposition, named after Camille Jordan and Claude Chevalley, expresses a linear operator in a unique way as the sum of two other linear operators which are simpler to understand. Specifically, one part is potentially diagonalisable and the other is nilpotent. The two parts are polynomials in the operator, which makes them behave nicely in algebraic manipulations.

The decomposition has a short description when the Jordan normal form of the operator is given, but it exists under weaker hypotheses than are needed for the existence of a Jordan normal form. Hence the Jordan–Chevalley decomposition can be seen as a generalisation of the Jordan normal form, which is also reflected in several proofs of it.

It is closely related to the Wedderburn principal theorem about associative algebras, which also leads to several analogues in Lie algebras. Analogues of the Jordan–Chevalley decomposition also exist for elements of Linear algebraic groups and Lie groups via a multiplicative reformulation. The decomposition is an important tool in the study of all of these objects, and was developed for this purpose.

In many texts, the potentially diagonalisable part is also characterised as the semisimple part.