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.
Introduction A basic question in linear algebra is whether an operator on a finite-dimensional vector space can be diagonalised. For example, this is closely related to the eigenvalues of the operator. In several contexts, one may be dealing with many operators which are not diagonalisable. Even over an algebraically closed field, a diagonalisation may not exist. In this context, the Jordan normal form achieves the best possible result akin to a diagonalisation. For linear operators over a field which is not algebraically closed, there may be no eigenvector at all. This latter point is not the main concern dealt with by the Jordan–Chevalley decomposition. To avoid this problem, instead potentially diagonalisable operators are considered, which are those that admit a diagonalisation over some field (or equivalently over the algebraic closure of the field under consideration). The operators which are "the furthest away" from being diagonalisable are nilpotent operators. An operator (or more generally an element of a ring) x {\displaystyle x} is said to be nilpotent when there is some positive integer m ≥ 1 {\displaystyle m\geq 1} such that x m = 0 {\displaystyle x^{m}=0} . In several contexts in abstract algebra, it is the case that the presence of nilpotent elements of a ring make them much more complicated to work with. To some extent, this is also the case for linear operators. The Jordan–Chevalley decomposition "separates out" the nilpotent part of an operator which causes it to be not potentially diagonalisable. So when it exists, the complications introduced by nilpotent operators and their interaction with other operators can be understood using the Jordan–Chevalley decomposition. Historically, the Jordan–Chevalley decomposition was motivated by the applications to the theory of Lie algebras and linear algebraic groups, as described in sections below.
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