Jordan–Chevalley decomposition

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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

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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)   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.[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. 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,[1] as described in sections below.

Decomposition of a linear operator

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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[citation needed], but can be generalised to cases where the minimal polynomial of   does not split.

If the minimal polynomial of   splits into 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  .

Jordan-Chevalley decomposition — Let   be any operator on the finite-dimensional vector space   over the field  . Then   admits a Jordan-Chevalley decomposition if and only if the minimal polynomial of   is a product of separable polynomials. Moreover, in this case, there is a unique Jordan-Chevalley decomposition, and   (and hence also  ) can be written as a polynomial (with coefficients from  ) in   with zero constant coefficient.

Several proofs are discussed in (Couty, Esterle & Zarouf 2011). Two arguments are also described below.

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.[2] (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.[citation needed] For these reasons, many texts restrict to the case of perfect fields.

Proof of uniqueness and necessity

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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 diagonalizable 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 separable 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 copies of  ), so is the minimal polynomial.

Concrete example for non-existence

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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.

Concrete argument for non-existence of a Jordan-Chavelley decomposition

Note that   has as its invariant  -linear subspaces precisely the ideals of   viewed as a ring, which correspond to the ideals of   containing  . Since   is irreducible in   ideals of   are     and   Suppose   for commuting  -linear operators   and   that are respectively semisimple (just over  , which is weaker than semisimplicity over an algebraic closure of   and also weaker than being potentially diagonalisable) and nilpotent. Since   and   commute, they each commute with   and hence each acts  -linearly on  . Therefore   and   are each given by multiplication by respective members of     and   with  . Since   is nilpotent,   is nilpotent in   therefore   in   for   is a field. Hence,   therefore   for some polynomial  . Also, we see that  . Since   is of characteristic   we have  . On the other hand, since   in   we have   therefore   in   Since   we have   Combining these results we get   This shows that   generates   as a  -algebra and thus the  -stable  -linear subspaces of   are ideals of   i.e. they are     and   We see that   is an  -invariant subspace of   which has no complement  -invariant subspace, contrary to the assumption that   is semisimple. Thus, there is no decomposition of   as a sum of commuting  -linear operators that are respectively semisimple and nilpotent.

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  .

Elementary proof of existence

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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. In this form, it is taken from (Geck 2022).

Let   have minimal polynomial   and assume this is a product of separable polynomials. This condition is equivalent to demanding that there is some separable   such that   and   for some  . By the Bézout lemma, there are polynomials   and   such that  . This can be used to define a recursion  , starting with  . Letting   be the algebra of operators which are polynomials in  , it can be checked by induction that for all  :

  •   because in each step, a polynomial is applied,
  •   because   for some   (by the algebraic version of Taylor's theorem). By definition of   as well as of   and  , this simplifies to  , which indeed lies in   by induction hypothesis,
  •   because   and both terms are in  , the first by the preceding point and the second by induction hypothesis.

Thus, as soon as  ,   by the second point since   and  , so the minimal polynomial of   will divide   and hence be separable. Moreover,   will be a polynomial in   by the first point and   will be nilpotent by the third point (in fact,  ). Therefore,   is then the Jordan–Chevalley decomposition of  . Q.E.D.

This proof, besides being completely elementary, has the advantage that it is algorithmic: By the Cayley–Hamilton theorem,   can be taken to be the characteristic polynomial of  , and in many contexts,   can be determined from  .[3] Then   can be determined using the Euclidean algorithm. The iteration of applying the polynomial   to the matrix then can be performed until either   (because then all later values will be equal) or   exceeds the dimension of the vector space on which   is defined (where   is the number of iteration steps performed, as above).

Proof of existence via Galois theory

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This proof, or variants of it, is commonly used to establish the Jordan–Chevalley decomposition. It has the advantage that it is very direct and describes quite precisely how close one can get to a Jordan–Chevalley decomposition: If   is the splitting field of the minimal polynomial of   and   is the group of automorphisms of   that fix the base field  , then the set   of elements of   that are fixed by all elements of   is a field with inclusions   (see Galois correspondence). Below it is argued that   admits a Jordan–Chevalley decomposition over  , but not any smaller field.[citation needed] This argument does not use Galois theory. However, Galois theory is required deduce from this the condition for the existence of the Jordan-Chevalley given above.

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.

Proof (Jordan-Chevalley decomposition from Jordan normal form)

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.

This fact can be used to deduce the Jordan–Chevalley decomposition in the general case. Let   be the splitting field of the minimal polynomial of  , so that   does admit a Jordan normal form over  . Then, by the argument just given,   has a Jordan–Chevalley decomposition   where   is a polynomial with coefficients from  ,   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  . Therefore, by definition,   are endomorphisms (represented by matrices) over  . Finally, since   contains an  -basis that spans the space containing  , by the same argument, we also see that   has coefficients in  . Q.E.D.

If the minimal polynomial of   is a product of separable polynomials, then the field extension   is Galois, meaning that  .

Relations to the theory of algebras

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Separable algebras

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The Jordan–Chevalley decomposition is very closely related to the Wedderburn principal theorem in the following formulation:[4]

Wedderburn principal theorem — Let   be a finite-dimensional associative algebra over the field   with Jacobson radical  . Then   is separable if and only if   has a separable semisimple subalgebra   such that  .

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.

Proof of equivalence between Wedderburn principal theorem and Jordan-Chevalley decomposition

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  , 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   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 of uniqueness and necessity, 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). The algebra   is separable and semisimple by assumption.

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 or having trivial radical, at least in the finite-dimensional case.)

Preservation under representations

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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 semisimple Lie algebra can be realised in this way.[6]

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:

  1. If  , then  , since   is the difference of the left and right multiplications by y.
  2. 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  ).[7]

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  .[8][9]

Nilpotency criterion

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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,

Nilpotency criterion — [10]  is nilpotent (i.e.,  ) if and only if   for every  . Also, if  , then it suffices the condition holds for   complex conjugation.

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:[11] 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

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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 = kan 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

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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

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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,[12] 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.[13]

Real semisimple Lie groups

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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.

References

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  1. ^ Couty, Esterle & Zarouf 2011, pp. 15–19
  2. ^ Conrad, Keith. "Semisimplicity" (PDF). Expository papers. Retrieved January 9, 2024.
  3. ^ Geck 2022, pp. 2–3
  4. ^ Ring Theory. Academic Press. 18 April 1972. ISBN 9780080873572.
  5. ^ Cohn, Paul M. (2002). Further Algebra and Applications. Springer London. ISBN 978-1-85233-667-7.
  6. ^ Humphreys 1972, p. 8
  7. ^ This is not easy to see in general but is shown in the proof of (Jacobson 1979, Ch. III, § 7, Theorem 11.). Editorial note: we need to add a discussion of this matter to "semisimple operator".
  8. ^ Weber, Brian (2 October 2012). "Lecture 8 - Preservation of the Jordan Decomposition and Levi's Theorem" (PDF). Course Notes. Retrieved 9 January 2024.
  9. ^ Fulton & Harris 1991, Theorem 9.20.
  10. ^ Serre 1992, LA 5.17. Lemma 6.7. The endomorphism
  11. ^ Serre 1992, LA 5.19. Theorem 7.1.
  12. ^ Waterhouse 1979, Theorem 9.2.
  13. ^ Waterhouse 1979, Theorem 9.3.