Smith normal form
E838595
Smith normal form is a canonical diagonal matrix form over the integers that classifies finitely generated abelian groups and simplifies solving systems of linear Diophantine equations.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Smith normal form canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10055754 — 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: Smith normal form Context triple: [Gröbner basis, generalizationOf, Smith normal form]
-
A.
Hermite normal form
Hermite normal form is a canonical matrix form used in linear algebra and number theory to uniquely represent integer matrices and solve systems of linear Diophantine equations.
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B.
Jordan normal form 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.
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C.
Sylvester matrix
The Sylvester matrix is a structured matrix constructed from the coefficients of two polynomials, commonly used to compute their resultant and study common roots in algebra.
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D.
Cayley–Hamilton theorem
The Cayley–Hamilton theorem is a fundamental result in linear algebra stating that every square matrix satisfies its own characteristic polynomial.
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E.
Bartels–Stewart algorithm
The Bartels–Stewart algorithm is a numerical linear algebra method that efficiently solves certain matrix equations, particularly Sylvester and Lyapunov equations, using Schur decompositions.
- 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: Smith normal form Target entity description: Smith normal form is a canonical diagonal matrix form over the integers that classifies finitely generated abelian groups and simplifies solving systems of linear Diophantine equations.
-
A.
Hermite normal form
Hermite normal form is a canonical matrix form used in linear algebra and number theory to uniquely represent integer matrices and solve systems of linear Diophantine equations.
-
B.
Jordan normal form 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.
-
C.
Sylvester matrix
The Sylvester matrix is a structured matrix constructed from the coefficients of two polynomials, commonly used to compute their resultant and study common roots in algebra.
-
D.
Cayley–Hamilton theorem
The Cayley–Hamilton theorem is a fundamental result in linear algebra stating that every square matrix satisfies its own characteristic polynomial.
-
E.
Bartels–Stewart algorithm
The Bartels–Stewart algorithm is a numerical linear algebra method that efficiently solves certain matrix equations, particularly Sylvester and Lyapunov equations, using Schur decompositions.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
canonical form
ⓘ
concept in abstract algebra ⓘ concept in linear algebra ⓘ matrix normal form ⓘ |
| appliesTo |
integer matrices
ⓘ
matrices over principal ideal domains ⓘ |
| assumes | ring is a principal ideal domain for full theory ⓘ |
| characterizedBy |
diagonal entries satisfying divisibility chain
ⓘ
each diagonal entry divides the next one ⓘ off-diagonal entries equal to zero ⓘ |
| definedOver |
integers
ⓘ
principal ideal domains ⓘ |
| encodes |
elementary divisors of a module over a PID
ⓘ
invariant factors of a finitely generated abelian group ⓘ |
| generalizes | diagonalization of integer matrices via unimodular transformations ⓘ |
| guarantees |
existence for any integer matrix
ⓘ
uniqueness up to multiplication by units ⓘ |
| hasAlternativeName | Smith canonical form NERFINISHED ⓘ |
| hasDiagonalEntries | invariant factors ⓘ |
| hasProperty |
canonical up to multiplication by units
ⓘ
diagonal matrix form ⓘ |
| implies | decomposition of finitely generated abelian groups into cyclic components ⓘ |
| introducedBy | Henry John Stephen Smith NERFINISHED ⓘ |
| involves |
unimodular column operations
ⓘ
unimodular row operations ⓘ |
| obtainedBy | left and right multiplication by unimodular matrices ⓘ |
| relatedTo |
Hermite normal form
NERFINISHED
ⓘ
Jordan normal form NERFINISHED ⓘ elementary divisor decomposition ⓘ finitely generated modules over a PID ⓘ invariant factor decomposition ⓘ rational canonical form NERFINISHED ⓘ |
| usedFor |
classification of finitely generated abelian groups
ⓘ
computing determinant up to units ⓘ computing elementary divisors ⓘ computing greatest common divisors of minors ⓘ computing invariant factors ⓘ computing invariants of modules over principal ideal domains ⓘ computing rank of an integer matrix ⓘ computing structure of abelian groups ⓘ simplifying integer linear systems ⓘ solving systems of linear Diophantine equations ⓘ |
| usedIn |
algebraic number theory
ⓘ
algebraic topology ⓘ coding theory ⓘ combinatorics ⓘ computational algebra ⓘ |
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: Smith normal form Description of subject: Smith normal form is a canonical diagonal matrix form over the integers that classifies finitely generated abelian groups and simplifies solving systems of linear Diophantine equations.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.