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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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 ⓘ

Referenced by (2)

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 ⓘ