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 many 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 to obtain something similar to a diagonalisation. For linear operators over a field which is not algebraically closed, there may be no eigenvector at all. The latter point is not the main concern in discussions on the Jordan-Chevalley decomposition. To avoid this problem, one can instead consider potentially diagonalisable operators, 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) 
is said to be nilpotent when there is some positive integer 
such that 
. 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 a large extent, this is also the case for linear operators. The remarkable fact is that one can „separate out“ the nilpotent part of an operator which causes it to be not potentially diagonalisable. This is what is described by 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.
Decomposition of a linear operator
Let 
be a field, 
a finite-dimensional vector space over , and 
a linear operator over (equivalently, a matrix with entries from ). If the minimal polynomial of splits over (for example if is algebraically closed), then has a Jordan normal form 
. If 
is the diagonal of 
, let 
be the remaining part. Then 
is a decomposition where 
is diagonalisable and 
is nilpotent. This restatement of the normal form as an additive decomposition not only makes the numerical computation more stable, but can be generalised to cases where the minimal polynomial of does not split.
If the minimal polynomial of splits into \textit{distinct} linear factors, then is diagonalisable. Therefore, if the minimal polynomial of is at least separable, then is potentially diagonalisable. The Jordan-Chevalley decomposition is concerned with the more general case where the minimal polynomial of is a product of separable polynomials.
Let 
be any linear operator on the finite-dimensional vector space over the field . A Jordan–Chevalley decomposition of is an expression of it as a sum
-
,
where 
is potentially diagonalisable, 
is nilpotent, and 
.
Several proofs are discussed in . A short direct version of the argument is given in .
If is a perfect field, then every polynomial is a product of separable polynomials (since every polynomial is a product of its irreducible factors, and these are separable over a perfect field). So in this case, the Jordan-Chevalley decomposition always exists. Moreover, over a perfect field, a polynomial is separable if and only if it is square-free. Therefore an operator is potentially diagonalisable if and only if its minimal polynomial is square-free. In general (over any field), the minimal polynomial of a linear operator is square-free if and only if the operator is semisimple. (In particular, the sum of two commuting semisimple operators is always semisimple over a perfect field. The same statement is not true over general fields.) 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. For these reasons, many texts restrict to the case of perfect fields.
Proof of uniqueness and necessity
That and are polynomials in implies in particular that they commute with any operator that commutes with . This observation underlies the uniqueness proof.
Let be a Jordan-Chevalley decomposition in which and (hence also) are polynomials in . Let 
be any Jordan-Chevalley decomposition. Then 
, and 
both commute with , hence with 
since these are polynomials in . The sum of commuting nilpotent operators is again nilpotent, and the sum of commuting potentially diagonalisable operators again potentially diagonalisable (because they are simultaneously diagonalisable over the algebraic closure of ). Since the only operator which is both potentially diagonalisable and nilpotent is the zero operator it follows that 
.
To show that the condition that have a minimal polynomial which is a product of separable polynomials is necessary, suppose that is some Jordan-Chevalley decomposition. Letting 
be the minimal polynomial of , one can check using the binomial theorem that 
can be written as 
where 
is some polynomial in . Moreover, for some 
, 
. Thus 
and so the minimal polynomial of must divide 
. As is a product of separable polynomials (namely of ), so is the minimal polynomial.
Concrete example for non-existence
If the ground field is not perfect, then a Jordan–Chevalley decomposition may not exist, as it is possible that the minimal polynomial is not a product of separable polynomials. The simplest such example is the following. Let be a prime number, let 
be an imperfect field of characteristic 
(e. g. 
) and choose 
that is not a th power. Let 
let 
be the image in the quotient and let be the -linear operator given by multiplication by in . Note that the minimal polynomial is precisely 
, which is inseparable and a square. By the necessity of the condition for the Jordan-Chevalley decomposition (as shown in the last section), this operator does not have a Jordan-Chevalley decomposition. It can be instructive to see concretely why there is at least no decomposition into a square-free and a nilpotent part.
If instead of with the polynomial , the same construction is performed with 
, the resulting operator still does not admit a Jordan-Chevalley decomposition by the main theorem. However, is semi-simple. The trivial decomposition 
hence expresses as a sum of a semisimple and a nilpotent operator, both of which are polynomials in .
Proof of existence via Galois theory
This is one of the most typical proofs of the Jordan Chevalley decomposition. It has the advantage that it describes quite precisely how close one can get to a Jordan Chevalley decomposition.
Above it was observed that if has a Jordan normal form (i. e. if the minimal polynomial of splits), then it has a Jordan Chevalley decomposition. In this case, one can also see directly that (and hence also ) is a polynomial in . Indeed, it suffices to check this for the decomposition of the Jordan matrix 
. This is a technical argument, but does not require any tricks beyond the Chinese remainder theorem.
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Proof (Jordan-Chevalley decomposition from Jordan normal form)
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In the Jordan normal form, we have written  where  is the number of Jordan blocks and  is one Jordan block. Now let  be the characteristic polynomial of . Because  splits, it can be written as  , where is the number of Jordan blocks,  are the distinct eigenvalues, and  are the sizes of the Jordan blocks, so  . Now, the Chinese remainder theorem applied to the polynomial ring  gives a polynomial  satisfying the conditions
-
 (for all i).
(There is a redundancy in the conditions if some is zero but that is not an issue; just remove it from the conditions.) The condition  , when spelled out, means that  for some polynomial  . Since  is the zero map on  ,  and agree on each ; i.e.,  . Also then  with  . The condition  ensures that and  have no constant terms. This completes the proof of the theorem in case the minimal polynomial of splits.
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This fact can be used to deduce the Jordan-Chevalley decomposition in the general case. Let have minimal polynomial and assume this is a product of separable polynomials. This condition is equivalent to demanding that there is some 
such that 
and 
for some . Let 
be the splitting field of , over which also splits. Therefore, by the argument just given, has a Jordan-Chevalley decomposition 
where 
is a polynomial with coefficients from , <c(x)> is diagonalisable (over ) and 
is nilpotent.
Let 
be a field automorphism of which fixes . Then
Here
is a polynomial in
, so is
. Thus,
and
commute. Also,
is potentially diagonalisable and
is nilpotent. Thus, by the uniqueness of the Jordan-Chevalley decomposition (over
),
and
. Because
is separable, the extension
is
Galois, meaning an element of
fixed by all automorphisms of
which fix
must lie in
. That is,
are endomorphisms (represented by matrices) over
. Finally, since
contains a
-basis that spans the space containing
, by the same argument, we also see that
has coefficients in
. This completes the proof.
Q.E.D.
This argument describes precisely why the condition on is required in the theorem and how close to a Jordan-Chevalley decomposition one can get in general: If is the minimal polynomial of , and is the splitting field of , then admits a Jordan-Chevalley decomposition over the subfield of of elements fixed by the entire Galois group 
(but not over smaller subfields).
Relations to the theory of algebras
Separable algebras
The Jordan-Chevalley decomposition is very closely related to the Wedderburn principal theorem in the following formulation:
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. 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.
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Proof of equivalence between Wedderburn principal theorem and Jordan-Chevalley decomposition
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To see how the Jordan-Chevalley decomposition follows from the Wedderburn principal theorem, let be a finite-dimensional vector space over the field , an endomorphism with a minimal polynomial which is a product of separable polynomials and  the subalgebra generated by . Note that  is a commutative Artinian ring, so is also the nilradical of . Moreover,  is separable, because if  , then for minimal polynomial , there is a separable polynomial such that and for some  . Therefore  , so the minimal polynomial of the image  divides , meaning that it must be separable as well (since a divisor of a separable polynomial is separable). There is then the vector-space decomposition  with  separable. In particular, the endomorphism can be written as where  and  . Moreover, both elements are, like any element of </math> A </math>, polynomials in .
Conversely, the Wedderburn principal theorem in the formulation above is a consequence of the Jordan-Chevalley decomposition. If has a separable subalgebra such that , then of course  is separable. Conversely, if is separable, then any element of is a sum of a separable and a nilpotent element. As shown above in #{proof for uniqueness}, this implies that the minimal polynomial will be a product of separable polynomials. Let  be arbitrary, define the operator  , and note that this has the same minimal polynomial as . So it admits a Jordan-Chevalley decomposition, where both operators are polynomials in  , hence of the form  for some  which have separable and nilpotent minimal polynomials, respectively. Moreover, this decomposition is unique. Thus if is the subalgebra of all separable elements (that this is a subalgebra can be seen by recalling that  is separable if and only if  is potentially diagonalisable), (because is the ideal of nilpotent elements).
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Over perfect fields, this result simplifies. Indeed, is then always separable in the sense of minimal polynomials: If , then the minimal polynomial is a product of separable polynomials, so there is a separable polynomial such that and for some . Thus . So in , the minimal polynomial of 
divides and is hence separable. The crucial point in the theorem is then not that is separable (because that condition is vacuous), but that it is semisimple, meaning its radical is trivial.
The same statement is true for Lie algebras, but only in characteristic zero. This is the content of Levi’s theorem. (Note that the notions of semisimple in both results do indeed correspond, because in both cases this is equivalent to being the sum of simple subalgebras, at least in the finite-dimensional case.)
Preservation under representations
The crucial point in the proof for the Wedderburn principal theorem above is that an element corresponds to a linear operator 
with the same properties. In the theory of Lie algebras, this corresponds to the adjoint representation of a Lie algebra 
. This decomposed operator has a Jordan-Chevalley decomposition 
. Just as in the associative case, this corresponds to a decomposition of , but polynomials are not available as a tool. One context in which this does makes sense is the restricted case where is contained in the Lie algebra 
of the endomorphisms of a finite-dimensional vector space over the perfect field . Indeed, any Lie algebra can be realised in this way.
If is the Jordan decomposition, then 
is the Jordan decomposition of the adjoint endomorphism 
on the vector space . Indeed, first, 
and 
commute since 
. Second, in general, for each endomorphism 
, we have:
- If
, then 
, since 
is the difference of the left and right multiplications by y.
- If is semisimple, then is semisimple, since semisimple is equivalent to potentially diagonalisable over a perfect field (if is diagonal over the basis
, then is diagonal over the basis consisting of the maps 
with 
and 
for 
).
Hence, by uniqueness, 
and 
.
The adjoint representation is a very natural and general representation of any Lie algebra. The argument above illustrates (and indeed proves) a general principle which generalises this: If 
is any finite-dimensional representation of a semisimple finite-dimensional Lie algebra over a perfect field, then 
preserves the Jordan decomposition in the following sense: if , then 
and 
.
Nilpotency criterion
The Jordan decomposition can be used to characterize nilpotency of an endomorphism. Let k be an algebraically closed field of characteristic zero, 
the endomorphism ring of k over rational numbers and V a finite-dimensional vector space over k. Given an endomorphism , let 
be the Jordan decomposition. Then is diagonalizable; i.e., 
where each is the eigenspace for eigenvalue with multiplicity 
. Then for any 
let 
be the endomorphism such that 
is the multiplication by 
. Chevalley calls 
the replica of given by 
. (For example, if 
, then the complex conjugate of an endomorphism is an example of a replica.) Now,
Proof: First, since 
is nilpotent,
-
.
If is the complex conjugation, this implies 
for every i. Otherwise, take to be a 
-linear functional 
followed by 
. Applying that to the above equation, one gets:
and, since are all real numbers, 
for every i. Varying the linear functionals then implies for every i. 
A typical application of the above criterion is the proof of Cartan's criterion for solvability of a Lie algebra. It says: if 
is a Lie subalgebra over a field k of characteristic zero such that 
for each 
, then is solvable.
Proof: Without loss of generality, assume k is algebraically closed. By Lie's theorem and Engel's theorem, it suffices to show for each 
, is a nilpotent endomorphism of V. Write 
. Then we need to show:
is zero. Let 
. Note we have: 
and, since 
is the semisimple part of the Jordan decomposition of 
, it follows that is a polynomial without constant term in ; hence, 
and the same is true with in place of . That is, 
, which implies the claim given the assumption.
Real semisimple Lie algebras
In the formulation of Chevalley and Mostow, the additive decomposition states that an element X in a real semisimple Lie algebra g with Iwasawa decomposition g = k ⊕ a ⊕ n can be written as the sum of three commuting elements of the Lie algebra X = S + D + N, with S, D and N conjugate to elements in k, a and n respectively. In general the terms in the Iwasawa decomposition do not commute.
Multiplicative decomposition
If is an invertible linear operator, it may be more convenient to use a multiplicative Jordan-Chevalley decomposition. This expresses as a product
-
,
where is potentially diagonalisable, and 
is nilpotent (one also says that 
is unipotent).
The multiplicative version of the decomposition follows from the additive one since, as is invertible (because the sum of an invertible operator and a nilpotent operator is invertible)
and 
is unipotent. (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. For this it is again useful to assume that the underlying field is perfect because then the Jordan-Chevalley decomposition exists for all matrices.
Linear algebraic groups
Let 
be a linear algebraic group over a perfect field. Then, essentially by definition, there is a closed embedding 
. Now, to each element 
, by the multiplicative Jordan decomposition, there are a pair of a semisimple element 
and a unipotent element 
a priori in 
such that 
. But, as it turns out, the elements 
can be shown to be in (i.e., they satisfy the defining equations of G) and that they are independent of the embedding into ; i.e., the decomposition is intrinsic.
When G is abelian, is then the direct product of the closed subgroup of the semisimple elements in G and that of unipotent elements.
Real semisimple Lie groups
The multiplicative decomposition states that if g is an element of the corresponding connected semisimple Lie group G with corresponding Iwasawa decomposition G = KAN, then g can be written as the product of three commuting elements g = sdu with s, d and u conjugate to elements of K, A and N respectively. In general the terms in the Iwasawa decomposition g = kan do not commute.
containing 
is irreducible in 
ideals of 
Suppose 
for commuting 
and 
that are respectively semisimple (just over 
-linearly on 
and 
with 
. Since 
is nilpotent in 
therefore 
in 
for 
is a field. Hence, 
therefore 
for some polynomial 
. Also, we see that 
. Since 
. On the other hand, since 
in 
we have 
therefore 
in 
Since 
we have 
Combining these results we get 
This shows that 
