Cauchy matrix
E239295
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Cauchy matrix canonical | 3 |
| Cauchy-like matrix | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2171657 — 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: Cauchy matrix Context triple: [Augustin-Louis Cauchy, knownFor, Cauchy matrix]
-
A.
Jacobi matrix
A Jacobi matrix is a tridiagonal matrix, often symmetric, that arises in numerical analysis and mathematical physics, particularly in the study of orthogonal polynomials and eigenvalue problems.
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B.
Cauchy-à-la-Tour
Cauchy-à-la-Tour is a small commune in the Pas-de-Calais department of northern France.
-
C.
Gaussian elimination
Gaussian elimination is a fundamental algorithm in linear algebra used to solve systems of linear equations by systematically transforming matrices into row-echelon form.
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D.
Kronecker product
The Kronecker product is a matrix operation that forms a large block matrix from two smaller matrices and is widely used in linear algebra, quantum computing, and signal processing.
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E.
Carathéodory–Fejér interpolation
Carathéodory–Fejér interpolation is a classical result in complex analysis and approximation theory that concerns constructing analytic functions, typically with bounded or positive real part, that match prescribed initial Taylor coefficients.
- 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: Cauchy matrix Target entity description: 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.
-
A.
Jacobi matrix
A Jacobi matrix is a tridiagonal matrix, often symmetric, that arises in numerical analysis and mathematical physics, particularly in the study of orthogonal polynomials and eigenvalue problems.
-
B.
Cauchy-à-la-Tour
Cauchy-à-la-Tour is a small commune in the Pas-de-Calais department of northern France.
-
C.
Gaussian elimination
Gaussian elimination is a fundamental algorithm in linear algebra used to solve systems of linear equations by systematically transforming matrices into row-echelon form.
-
D.
Kronecker product
The Kronecker product is a matrix operation that forms a large block matrix from two smaller matrices and is widely used in linear algebra, quantum computing, and signal processing.
-
E.
Carathéodory–Fejér interpolation
Carathéodory–Fejér interpolation is a classical result in complex analysis and approximation theory that concerns constructing analytic functions, typically with bounded or positive real part, that match prescribed initial Taylor coefficients.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
matrix class
ⓘ
structured matrix ⓘ |
| admits |
fast algorithms for computing determinants
ⓘ
fast algorithms for matrix-vector multiplication ⓘ fast algorithms for solving linear systems ⓘ |
| appearsIn |
barycentric form of Lagrange interpolation
ⓘ
partial fraction expansions ⓘ resolvent representations ⓘ |
| definedOver |
complex numbers
ⓘ
real numbers ⓘ |
| dependsOn |
sequence (x_i)
ⓘ
sequence (y_j) ⓘ |
| generalizedBy |
Cauchy matrix
self-linksurface differs
ⓘ
surface form:
Cauchy-like matrix
|
| hasDeterminantFormula | det(C) = (∏_{i<k}(x_i - x_k) ∏_{j<ℓ}(y_ℓ - y_j)) / (∏_{i,j}(x_i - y_j)) ⓘ |
| hasEntryFormula | a_{ij} = 1 / (x_i - y_j) ⓘ |
| hasProperty |
all entries are rational functions of x_i and y_j
ⓘ
can be scaled to improve conditioning ⓘ determinant has closed-form product expression ⓘ entries are analytic functions of parameters where defined ⓘ entries are reciprocals of pairwise differences ⓘ examples are ill-conditioned for certain node choices ⓘ inverse has explicit formula ⓘ low displacement rank ⓘ nonsingular if x_i and y_j are pairwise distinct and disjoint ⓘ rank equals matrix size when nonsingular ⓘ structure can be exploited in numerical linear algebra libraries ⓘ |
| hasSpecialCase |
complex Cauchy matrix
ⓘ
real Cauchy matrix ⓘ symmetric Cauchy matrix ⓘ |
| namedAfter | Augustin-Louis Cauchy ⓘ |
| relatedTo |
Hankel matrix
ⓘ
Toeplitz matrix ⓘ Vandermonde matrix ⓘ displacement structure ⓘ |
| requiresCondition | x_i - y_j ≠ 0 for all i,j ⓘ |
| usedIn |
algebra
ⓘ
approximation theory ⓘ barycentric interpolation formulas ⓘ coding theory ⓘ moment problems ⓘ numerical analysis ⓘ polynomial interpolation ⓘ rational interpolation ⓘ signal processing ⓘ spectral methods ⓘ 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: Cauchy matrix Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Cauchy-like matrix