Sylvester matrix
E571002
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Sylvester matrix canonical | 3 |
How this entity was disambiguated
This entity first appeared as the object of triple T6149920 — 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: Sylvester matrix Context triple: [James Joseph Sylvester, notableWork, Sylvester matrix]
-
A.
Cauchy matrix
A Cauchy matrix is a structured matrix whose entries are defined by the reciprocals of pairwise differences of two sequences, widely used in numerical analysis, interpolation, and algebra.
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B.
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.
-
C.
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.
-
D.
Cauchy determinant
The Cauchy determinant is a classical determinant formula in linear algebra that gives a closed-form expression for matrices with entries of the form 1/(x_i + y_j), named after the French mathematician Augustin-Louis Cauchy.
-
E.
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.
- 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: Sylvester matrix Target entity description: 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.
-
A.
Cauchy matrix
A Cauchy matrix is a structured matrix whose entries are defined by the reciprocals of pairwise differences of two sequences, widely used in numerical analysis, interpolation, and algebra.
-
B.
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.
-
C.
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.
-
D.
Cauchy determinant
The Cauchy determinant is a classical determinant formula in linear algebra that gives a closed-form expression for matrices with entries of the form 1/(x_i + y_j), named after the French mathematician Augustin-Louis Cauchy.
-
E.
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.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical object
ⓘ
matrix ⓘ structured matrix ⓘ |
| appearsIn |
eigenvalue methods for polynomial equations
ⓘ
resultant-based polynomial system solvers ⓘ |
| appliesTo |
multivariate polynomials via specialization
ⓘ
univariate polynomials ⓘ |
| constructionMethod |
rows formed from shifted coefficient vectors of the first polynomial
ⓘ
rows formed from shifted coefficient vectors of the second polynomial ⓘ |
| countryOfOrigin | United Kingdom ⓘ |
| dimensionRule | for degrees m and n the Sylvester matrix is of size (m+n)×(m+n) ⓘ |
| field |
algebra
ⓘ
linear algebra ⓘ |
| generalization |
block Sylvester matrix
ⓘ
multivariate Sylvester matrix ⓘ |
| hasProperty |
block Toeplitz structure
ⓘ
can be ill-conditioned for high-degree polynomials ⓘ depends linearly on polynomial coefficients ⓘ determinant equals the resultant of the polynomials ⓘ rectangular in general ⓘ square when constructed from two univariate polynomials of fixed degrees ⓘ |
| implies | vanishing determinant if and only if polynomials have a common root ⓘ |
| namedAfter | James Joseph Sylvester ⓘ |
| relatedTo |
Bezout matrix
ⓘ
Sylvester equation ⓘ Sylvester resultant ⓘ
linked to:
Sylvester determinant
companion matrix ⓘ discriminant of a polynomial ⓘ resultant ⓘ |
| symbol | often denoted S(f,g) for polynomials f and g ⓘ |
| typicalInput | coefficients of two polynomials ⓘ |
| usedFor |
computing polynomial greatest common divisors
ⓘ
computing the resultant of two polynomials ⓘ elimination theory ⓘ resultant-based root finding ⓘ studying common roots of two polynomials ⓘ testing whether two polynomials have a common root ⓘ |
| usedIn |
algebraic geometry
ⓘ
computational algebra ⓘ computer algebra systems ⓘ control theory ⓘ numerical linear algebra ⓘ signal processing ⓘ symbolic computation ⓘ systems theory ⓘ |
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: Sylvester matrix Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.