Hermite normal form
E502190
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Hermite normal form canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T5191843 — 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: Hermite normal form Context triple: [Charles Hermite, knownFor, Hermite normal form]
-
A.
Buchberger algorithm
The Buchberger algorithm is a fundamental procedure in computational algebra for computing Gröbner bases of polynomial ideals, enabling systematic solutions to systems of polynomial equations.
-
B.
Gröbner basis
A Gröbner basis is a particular generating set of an ideal in a polynomial ring that allows algorithmic solutions to many problems in computational algebra, such as ideal membership and solving systems of polynomial equations.
-
C.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
-
D.
Hilbert’s tenth problem
Hilbert’s tenth problem is a famous unsolved question in mathematics that asked for a general algorithm to determine whether any given Diophantine equation has an integer solution, and whose negative answer helped establish fundamental limits of computability.
-
E.
Hilbert’s fourteenth problem
Hilbert’s fourteenth problem is one of David Hilbert’s famous list of 23 problems, concerning the finite generation of certain algebras of invariants in algebraic geometry and invariant theory.
- 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: Hermite normal form Target entity description: 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.
-
A.
Buchberger algorithm
The Buchberger algorithm is a fundamental procedure in computational algebra for computing Gröbner bases of polynomial ideals, enabling systematic solutions to systems of polynomial equations.
-
B.
Gröbner basis
A Gröbner basis is a particular generating set of an ideal in a polynomial ring that allows algorithmic solutions to many problems in computational algebra, such as ideal membership and solving systems of polynomial equations.
-
C.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
-
D.
Hilbert’s tenth problem
Hilbert’s tenth problem is a famous unsolved question in mathematics that asked for a general algorithm to determine whether any given Diophantine equation has an integer solution, and whose negative answer helped establish fundamental limits of computability.
-
E.
Hilbert’s fourteenth problem
Hilbert’s fourteenth problem is one of David Hilbert’s famous list of 23 problems, concerning the finite generation of certain algebras of invariants in algebraic geometry and invariant theory.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
concept in linear algebra
ⓘ
concept in number theory ⓘ matrix normal form ⓘ |
| algorithmicProperty |
can be computed in polynomial time in the size of the input matrix
ⓘ
can be computed using elementary integer column operations ⓘ can be computed using elementary integer row operations ⓘ |
| alsoKnownAs | HNF NERFINISHED ⓘ |
| application |
computing bases of solution spaces of homogeneous Diophantine systems
ⓘ
computing integer hulls in polyhedral theory ⓘ cryptographic lattice constructions ⓘ solving Ax = b over the integers ⓘ |
| belongsTo | theory of finitely generated abelian groups ⓘ |
| canonicalFormFor |
integer matrices under left multiplication by unimodular matrices (row HNF)
ⓘ
integer matrices under right multiplication by unimodular matrices (column HNF) ⓘ |
| definedOver | integers ⓘ |
| diagonalCondition | diagonal entries are positive ⓘ |
| entryCondition | all entries are integers ⓘ |
| equivalenceRelation | two integer matrices are equivalent if they have the same Hermite normal form ⓘ |
| fieldRestriction | not defined as a normal form over arbitrary fields ⓘ |
| generalizationOf | Gaussian elimination to integer matrices with remainder constraints ⓘ |
| guarantees |
existence of a basis of the integer column space in triangular form
ⓘ
existence of a basis of the integer row space in triangular form ⓘ |
| matrixType |
lower triangular matrix (row HNF convention)
ⓘ
upper triangular matrix (column HNF convention) ⓘ |
| namedAfter | Charles Hermite NERFINISHED ⓘ |
| normalizationDirection |
column Hermite normal form
ⓘ
row Hermite normal form ⓘ |
| offDiagonalCondition |
entries above the diagonal are nonnegative (column HNF)
ⓘ
entries below the diagonal are nonnegative (row HNF) ⓘ |
| relatedTo |
Smith normal form
NERFINISHED
ⓘ
integer kernel computation ⓘ lattice basis reduction ⓘ unimodular matrix ⓘ |
| remainderCondition | off-diagonal entries are strictly smaller than the corresponding diagonal entry ⓘ |
| stabilityProperty | invariant under multiplication by unimodular matrices on one side ⓘ |
| uniquenessProperty | each integer matrix has a unique Hermite normal form up to unimodular transformations ⓘ |
| usedFor |
canonical representation of integer matrices
ⓘ
computing a basis of the integer column space ⓘ computing a basis of the integer row space ⓘ computing determinant of an integer matrix (up to sign) ⓘ computing lattice bases ⓘ computing rank of an integer matrix ⓘ solving systems of linear Diophantine equations ⓘ testing integer matrix equivalence ⓘ |
| usedIn |
algorithmic number theory
ⓘ
computational geometry of numbers ⓘ integer linear 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: Hermite normal form Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.