Jordan normal form theorem

E621088

mathematical theorem

The Jordan normal form theorem is a fundamental result in linear algebra that states every square matrix over an algebraically closed field is similar to a block diagonal matrix composed of Jordan blocks, providing a canonical form for linear operators.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

All labels observed (6)

Label	Occurrences
Jordan normal form	6
Jordan canonical form	2
Jordan decomposition	1
Jordan matrix	1
Jordan normal form theorem canonical	1
real Jordan form	1

How this entity was disambiguated

This entity first appeared as the object of triple T6832965 — 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 normal form theorem
Context triple: [linear algebra, hasKeyTheorem, Jordan normal form theorem]

A. Sylvester’s law of inertia
Sylvester’s law of inertia is a theorem in linear algebra stating that the numbers of positive, negative, and zero eigenvalues (the inertia) of a real symmetric matrix are invariant under change of basis.
B. Noether normalization lemma
The Noether normalization lemma is a fundamental result in commutative algebra and algebraic geometry that shows any finitely generated algebra over a field can be made integral over a polynomial subring, providing a way to relate complicated varieties to affine space.
C. Jordan–Hölder theorem
The Jordan–Hölder theorem is a fundamental result in group theory stating that any two composition series of a finite group have the same length and the same (up to order and isomorphism) simple factor groups.
D. Peter–Weyl theorem
The Peter–Weyl theorem is a fundamental result in representation theory and harmonic analysis that decomposes square-integrable functions on a compact topological group into a direct sum of finite-dimensional irreducible unitary representations.
E. Banach–Mazur theorem
The Banach–Mazur theorem is a fundamental result in functional analysis that characterizes separable Banach spaces as isometrically isomorphic to closed subspaces of spaces of continuous functions on compact metric spaces.
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 normal form theorem
Target entity description: The Jordan normal form theorem is a fundamental result in linear algebra that states every square matrix over an algebraically closed field is similar to a block diagonal matrix composed of Jordan blocks, providing a canonical form for linear operators.

A. Sylvester’s law of inertia
Sylvester’s law of inertia is a theorem in linear algebra stating that the numbers of positive, negative, and zero eigenvalues (the inertia) of a real symmetric matrix are invariant under change of basis.
B. Noether normalization lemma
The Noether normalization lemma is a fundamental result in commutative algebra and algebraic geometry that shows any finitely generated algebra over a field can be made integral over a polynomial subring, providing a way to relate complicated varieties to affine space.
C. Jordan–Hölder theorem
The Jordan–Hölder theorem is a fundamental result in group theory stating that any two composition series of a finite group have the same length and the same (up to order and isomorphism) simple factor groups.
D. Peter–Weyl theorem
The Peter–Weyl theorem is a fundamental result in representation theory and harmonic analysis that decomposes square-integrable functions on a compact topological group into a direct sum of finite-dimensional irreducible unitary representations.
E. Banach–Mazur theorem
The Banach–Mazur theorem is a fundamental result in functional analysis that characterizes separable Banach spaces as isometrically isomorphic to closed subspaces of spaces of continuous functions on compact metric spaces.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	mathematical theorem ⓘ
appliesTo	linear operators on finite-dimensional vector spaces ⓘ square matrices ⓘ
assumes	algebraically closed field ⓘ
concerns	decomposition of vector spaces into generalized eigenspaces ⓘ structure of linear operators ⓘ
concludes	every linear operator on a finite-dimensional vector space over an algebraically closed field has a Jordan normal form ⓘ every square matrix is similar to a block diagonal matrix of Jordan blocks ⓘ every square matrix over an algebraically closed field is similar to a Jordan matrix ⓘ
describes	Jordan normal form ⓘ linked to: Jordan normal form theorem
equivalentTo	classification of finite-dimensional modules over k[x] where k is algebraically closed ⓘ
field	linear algebra ⓘ matrix theory ⓘ representation theory ⓘ
hasComponentConcept	Jordan block ⓘ Jordan chain ⓘ Jordan matrix ⓘ
hasCondition	field must be algebraically closed for full Jordan form ⓘ
hasVariant	Jordan–Chevalley decomposition ⓘ real Jordan form ⓘ linked to: Jordan normal form theorem
historicalAttribution	Camille Jordan proved the theorem in the 19th century ⓘ
implies	every linear operator decomposes into semisimple and nilpotent parts that commute ⓘ existence of a basis of generalized eigenvectors ⓘ
involves	diagonal entries equal to eigenvalues ⓘ nilpotent Jordan blocks ⓘ superdiagonal entries equal to 1 in each Jordan block ⓘ upper triangular matrices ⓘ
namedAfter	Camille Jordan ⓘ
provides	canonical form for linear operators up to similarity ⓘ canonical form for matrices up to similarity ⓘ
relatesTo	characteristic polynomial ⓘ eigenvalues ⓘ eigenvectors ⓘ generalized eigenvectors ⓘ minimal polynomial ⓘ nilpotent operators ⓘ primary decomposition theorem ⓘ rational canonical form ⓘ similarity of matrices ⓘ
requires	finite-dimensional vector space ⓘ
typicalField	algebraic closure of a given field ⓘ complex numbers ⓘ
usedFor	analyzing dynamical systems ⓘ classification of linear operators up to similarity ⓘ computing matrix functions ⓘ solving systems of linear differential equations ⓘ studying representations of linear transformations ⓘ

How these facts were elicited