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.
The decomposition is an important tool in the study of all of these objects, and was developed for this purpose.
A basic question in linear algebra is whether an operator on a finite-dimensional vector space can be diagonalised.
In this context, the Jordan normal form achieves the best possible result akin to a diagonalisation.
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).
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.
[citation needed] 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.
This restatement of the normal form as an additive decomposition not only makes the numerical computation more stable[citation needed], but can be generalised to cases where the minimal polynomial of
The property of being semisimple is more relevant than being potentially diagonalisable in most contexts where the Jordan–Chevalley decomposition is applied, such as for Lie algebras.
[citation needed] For these reasons, many texts restrict to the case of perfect fields.
The sum of commuting nilpotent operators is again nilpotent, and the sum of commuting potentially diagonalisable operators again potentially diagonalisable (because they are simultaneously diagonalizable over the algebraic closure of
This construction is similar to Hensel's lemma in that it uses an algebraic analogue of Taylor's theorem to find an element with a certain algebraic property via a variant of Newton's method.
This proof, besides being completely elementary, has the advantage that it is algorithmic: By the Cayley–Hamilton theorem,
It has the advantage that it is very direct and describes quite precisely how close one can get to a Jordan–Chevalley decomposition: If
However, Galois theory is required deduce from this the condition for the existence of the Jordan-Chevalley given above.
This is a technical argument, but does not require any tricks beyond the Chinese remainder theorem.
Now, the Chinese remainder theorem applied to the polynomial ring
Usually, the term „separable“ in this theorem refers to the general concept of a separable algebra and the theorem might then be established as a corollary of a more general high-powered result.
[5] However, if it is instead interpreted in the more basic sense that every element have a separable minimal polynomial, then this statement is essentially equivalent to the Jordan–Chevalley decomposition as described above.
This gives a different way to view the decomposition, and for instance (Jacobson 1979) takes this route for establishing it.
Conversely, the Wedderburn principal theorem in the formulation above is a consequence of the Jordan–Chevalley decomposition.
is separable (because that condition is vacuous), but that it is semisimple, meaning its radical is trivial.
The crucial point in the proof for the Wedderburn principal theorem above is that an element
The argument above illustrates (and indeed proves) a general principle which generalises this: If
Proof:[11] Without loss of generality, assume k is algebraically closed.
(Conversely, by the same type of argument, one can deduce the additive version from the multiplicative one.)
The multiplicative version is closely related to decompositions encountered in a linear algebraic group.