Jordan normal form theorem
E621088
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.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Jordan canonical form | 1 |
| Jordan decomposition | 1 |
| Jordan matrix | 1 |
| Jordan normal form | 1 |
| Jordan normal form theorem canonical | 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]
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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.
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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.
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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.
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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.
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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 NERFINISHED ⓘ |
| equivalentTo | classification of finite-dimensional modules over k[x] where k is algebraically closed ⓘ |
| field |
linear algebra
ⓘ
matrix theory ⓘ representation theory ⓘ |
| hasComponentConcept |
Jordan block
NERFINISHED
ⓘ
Jordan chain ⓘ Jordan matrix NERFINISHED ⓘ |
| hasCondition | field must be algebraically closed for full Jordan form ⓘ |
| hasVariant |
Jordan–Chevalley decomposition
NERFINISHED
ⓘ
real Jordan form NERFINISHED ⓘ |
| historicalAttribution | Camille Jordan proved the theorem in the 19th century NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ 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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Jordan normal form theorem Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Jordan normal form
this entity surface form:
Jordan decomposition
this entity surface form:
Jordan matrix
this entity surface form:
Jordan canonical form