Jordan–Chevalley decomposition

E904578

concept in linear algebra concept in representation theory mathematical theorem

The Jordan–Chevalley decomposition is a fundamental result in linear algebra and representation theory that expresses a linear operator (or matrix) as the sum or product of commuting semisimple and nilpotent parts.

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All labels observed (3)

Label	Occurrences
Jordan–Chevalley decomposition canonical	3
Jordan decomposition	1
additive Jordan decomposition	1

How this entity was disambiguated

This entity first appeared as the object of triple T11099203 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Jordan–Chevalley decomposition
Context triple: [Camille Jordan, knownFor, Jordan–Chevalley decomposition]

A. Cartan decomposition
Cartan decomposition is a fundamental structural result in Lie theory that expresses a Lie algebra or Lie group as a direct sum or product of subspaces or subgroups with specific symmetry properties, widely used in differential geometry and representation theory.
B. Chevalley
Chevalley is a French surname most prominently associated with Claude Chevalley, a influential 20th-century mathematician known for his work in algebra and group theory.
C. Schur–Weyl duality
Schur–Weyl duality is a fundamental result in representation theory that links representations of the symmetric group and the general linear group via their commuting actions on tensor powers of a vector space.
D. Gelfand–Kirillov dimension
The Gelfand–Kirillov dimension is an invariant in noncommutative algebra that measures the growth rate of algebras and modules, serving as an analogue of Krull dimension for noncommutative settings.
E. Iwasawa decomposition
The Iwasawa decomposition is a fundamental factorization in Lie group theory that expresses a semisimple Lie group as a product of a maximal compact subgroup, a maximal abelian subgroup, and a nilpotent subgroup, playing a key role in representation theory and harmonic analysis.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Jordan–Chevalley decomposition
Target entity description: The Jordan–Chevalley decomposition is a fundamental result in linear algebra and representation theory that expresses a linear operator (or matrix) as the sum or product of commuting semisimple and nilpotent parts.

A. Cartan decomposition
Cartan decomposition is a fundamental structural result in Lie theory that expresses a Lie algebra or Lie group as a direct sum or product of subspaces or subgroups with specific symmetry properties, widely used in differential geometry and representation theory.
B. Chevalley
Chevalley is a French surname most prominently associated with Claude Chevalley, a influential 20th-century mathematician known for his work in algebra and group theory.
C. Schur–Weyl duality
Schur–Weyl duality is a fundamental result in representation theory that links representations of the symmetric group and the general linear group via their commuting actions on tensor powers of a vector space.
D. Gelfand–Kirillov dimension
The Gelfand–Kirillov dimension is an invariant in noncommutative algebra that measures the growth rate of algebras and modules, serving as an analogue of Krull dimension for noncommutative settings.
E. Iwasawa decomposition
The Iwasawa decomposition is a fundamental factorization in Lie group theory that expresses a semisimple Lie group as a product of a maximal compact subgroup, a maximal abelian subgroup, and a nilpotent subgroup, playing a key role in representation theory and harmonic analysis.
F. None of above. chosen

Statements (48)

Predicate	Object
instanceOf	concept in linear algebra ⓘ concept in representation theory ⓘ mathematical theorem ⓘ
alsoKnownAs	Jordan decomposition ⓘ linked to: Jordan–Chevalley decomposition additive Jordan decomposition ⓘ linked to: Jordan–Chevalley decomposition
appliesTo	linear operators ⓘ square matrices ⓘ
assumption	base field often assumed algebraically closed ⓘ characteristic of the field often assumed to be zero or sufficiently large ⓘ
component	nilpotent part ⓘ semisimple part ⓘ unipotent part ⓘ
context	finite-dimensional vector spaces ⓘ
describes	decomposition of a linear operator into semisimple and nilpotent parts ⓘ
field	linear algebra ⓘ representation theory ⓘ
generalizes	Jordan normal form for matrices ⓘ
hasForm	additive decomposition ⓘ multiplicative decomposition ⓘ
historicalPeriod	20th century mathematics ⓘ
holdsIn	any finite-dimensional representation of a Lie algebra over an algebraically closed field of characteristic zero ⓘ any finite-dimensional representation of a linear algebraic group over an algebraically closed field of characteristic zero ⓘ
implies	eigenvalues of the operator are eigenvalues of its semisimple part ⓘ nilpotent part has only zero as eigenvalue ⓘ
influencedBy	Jordan normal form ⓘ linked to: Jordan normal form theorem
influences	modern representation theory of Lie algebras ⓘ structure theory of linear algebraic groups ⓘ
namedAfter	Camille Jordan ⓘ Claude Chevalley ⓘ
over	algebraically closed field of characteristic zero ⓘ
property	decomposition is unique ⓘ semisimple and nilpotent parts are polynomials in the original operator ⓘ semisimple and nilpotent parts commute ⓘ
relatedConcept	nilpotent operator ⓘ primary decomposition theorem ⓘ semisimple operator ⓘ spectral decomposition ⓘ unipotent operator ⓘ
requires	minimal polynomial factorization into distinct and repeated irreducible factors ⓘ
statement	every invertible linear operator can be written as the product of a semisimple operator and a unipotent operator that commute ⓘ every linear operator can be written as the sum of a semisimple operator and a nilpotent operator that commute ⓘ
toolFor	analyzing representations via semisimple and nilpotent elements ⓘ defining semisimple and unipotent elements in algebraic groups ⓘ
usedIn	Jordan normal form ⓘ linked to: Jordan normal form theorem rational canonical form ⓘ representation theory of Lie algebras ⓘ representation theory of algebraic groups ⓘ structure theory of linear operators ⓘ

How these facts were elicited

Referenced by (5)

Full triples — surface form annotated when it differs from this entity's canonical label.

Camille Jordan → knownFor → Jordan–Chevalley decomposition ⓘ

Fitting lemma → relatedTo → Jordan–Chevalley decomposition ⓘ

Jordan normal form theorem → hasVariant → Jordan–Chevalley decomposition ⓘ

Jordan–Chevalley decomposition → alsoKnownAs → Jordan decomposition ⓘ

linked to: Jordan–Chevalley decomposition

Jordan–Chevalley decomposition → alsoKnownAs → additive Jordan decomposition ⓘ