Vandermonde matrix
E620657
A Vandermonde matrix is a structured matrix whose rows (or columns) are geometric progressions of given numbers, widely used in polynomial interpolation, determinant theory, and numerical analysis.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Vandermonde determinant | 5 |
| Vandermonde matrix canonical | 5 |
| Vandermonde polynomial | 1 |
| block Vandermonde matrix | 1 |
| confluent Vandermonde matrix | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6800911 — 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: Vandermonde matrix Context triple: [Lagrange interpolation polynomial, relatedTo, Vandermonde 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.
-
B.
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.
-
C.
Vandermonde's identity
Vandermonde's identity is a fundamental combinatorial formula that expresses a binomial coefficient with a sum index as a sum of products of binomial coefficients, often visualized via counting arguments or generating functions.
-
D.
Sylvester determinant
The Sylvester determinant is a mathematical construct introduced by James Joseph Sylvester, typically referring to a determinant associated with resultants and elimination theory in algebra.
-
E.
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.
- 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: Vandermonde matrix Target entity description: A Vandermonde matrix is a structured matrix whose rows (or columns) are geometric progressions of given numbers, widely used in polynomial interpolation, determinant theory, and numerical analysis.
-
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.
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.
-
C.
Vandermonde's identity
Vandermonde's identity is a fundamental combinatorial formula that expresses a binomial coefficient with a sum index as a sum of products of binomial coefficients, often visualized via counting arguments or generating functions.
-
D.
Sylvester determinant
The Sylvester determinant is a mathematical construct introduced by James Joseph Sylvester, typically referring to a determinant associated with resultants and elimination theory in algebra.
-
E.
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.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
matrix
ⓘ
structured matrix ⓘ |
| appearsIn |
Prony’s method
ⓘ
barycentric interpolation formulas ⓘ exponential fitting ⓘ moment problems ⓘ |
| belongsTo | class of polynomial evaluation matrices ⓘ |
| field |
determinant theory
ⓘ
linear algebra ⓘ numerical analysis ⓘ polynomial interpolation ⓘ |
| hasAlgorithm |
divide-and-conquer methods for Vandermonde systems
ⓘ
specialized O(n^2) algorithms for solving Vandermonde systems ⓘ |
| hasAlternativeBasis | can be replaced by orthogonal polynomial bases for better conditioning ⓘ |
| hasColumnForm | (1, x_j, x_j^2, …, x_j^{m-1})^T for given scalars x_j ⓘ |
| hasComputationIssue | direct solution is numerically unstable for large systems ⓘ |
| hasDeterminantFormula | det(V) = ∏_{1 ≤ i < j ≤ n} (x_j − x_i) ⓘ |
| hasGeneralization |
block Vandermonde matrix
ⓘ
linked to:
Vandermonde matrix
confluent Vandermonde matrix ⓘ
linked to:
Vandermonde matrix
q-Vandermonde matrix ⓘ |
| hasProperty |
can be expressed as evaluation matrix of monomials at nodes
ⓘ
columns form powers of the nodes x_i ⓘ determinant is zero iff some x_i = x_j for i ≠ j ⓘ ill-conditioned for large n or clustered nodes ⓘ rows form geometric progressions ⓘ |
| hasRowForm | (1, x_i, x_i^2, …, x_i^{n-1}) for given scalars x_i ⓘ |
| hasSize | m×n for m rows and n columns ⓘ |
| hasSpecialCase | DFT matrix when nodes are roots of unity ⓘ |
| isDefinedOver |
any field
ⓘ
complex numbers ⓘ real numbers ⓘ |
| isNonsingularIf | all x_i are pairwise distinct ⓘ |
| isSingularIf | two or more x_i coincide ⓘ |
| namedAfter | Alexandre-Théophile Vandermonde ⓘ |
| relatedTo |
Cauchy matrix
ⓘ
Lagrange interpolation ⓘ
linked to:
Lagrange interpolation polynomial
Newton interpolation ⓘ
linked to:
Newton interpolation polynomial
Toeplitz matrix ⓘ
linked to:
Toeplitz matrices
companion matrix ⓘ |
| usedFor |
coding theory
ⓘ
computing coefficients of interpolating polynomials ⓘ determinant evaluation ⓘ discrete Fourier transform generalizations ⓘ polynomial interpolation ⓘ signal processing ⓘ solving systems with polynomial data ⓘ system identification ⓘ |
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: Vandermonde matrix Description of subject: A Vandermonde matrix is a structured matrix whose rows (or columns) are geometric progressions of given numbers, widely used in polynomial interpolation, determinant theory, and numerical analysis.
Referenced by (13)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Vandermonde matrix
linked to: Vandermonde matrix
linked to: Vandermonde matrix
linked to: Vandermonde matrix
linked to: Vandermonde matrix
linked to: Vandermonde matrix
linked to: Vandermonde matrix
linked to: Vandermonde matrix