Jordan–Chevalley decomposition
E904578
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Jordan–Chevalley decomposition canonical | 2 |
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.
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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.
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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.
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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
NERFINISHED
ⓘ
additive Jordan decomposition NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| influences |
modern representation theory of Lie algebras
ⓘ
structure theory of linear algebraic groups ⓘ |
| namedAfter |
Camille Jordan
NERFINISHED
ⓘ
Claude Chevalley NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
rational canonical form ⓘ representation theory of Lie algebras ⓘ representation theory of algebraic groups ⓘ structure theory of linear operators ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
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Input
Subject: Jordan–Chevalley decomposition Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.