Cauchy determinant
E239296
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Cauchy determinant canonical | 1 |
| determinant de Cauchy | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2171658 — 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 determinant Context triple: [Augustin-Louis Cauchy, knownFor, Cauchy determinant]
-
A.
Cauchy-à-la-Tour
Cauchy-à-la-Tour is a small commune in the Pas-de-Calais department of northern France.
-
B.
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.
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C.
Jacobi polynomials
Jacobi polynomials are a family of classical orthogonal polynomials depending on two parameters, widely used in approximation theory, numerical analysis, and solutions of differential equations.
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D.
Jacobi triple product
The Jacobi triple product is a fundamental identity in number theory and complex analysis that expresses an infinite product as an infinite sum, playing a key role in the theory of theta functions and q-series.
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E.
Cauchy–Kovalevskaya theorem
The Cauchy–Kovalevskaya theorem is a fundamental result in partial differential equations that guarantees the existence and uniqueness of analytic solutions to certain initial value problems under appropriate analyticity conditions.
- 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 determinant Target entity description: 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.
-
A.
Cauchy-à-la-Tour
Cauchy-à-la-Tour is a small commune in the Pas-de-Calais department of northern France.
-
B.
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.
-
C.
Jacobi polynomials
Jacobi polynomials are a family of classical orthogonal polynomials depending on two parameters, widely used in approximation theory, numerical analysis, and solutions of differential equations.
-
D.
Jacobi triple product
The Jacobi triple product is a fundamental identity in number theory and complex analysis that expresses an infinite product as an infinite sum, playing a key role in the theory of theta functions and q-series.
-
E.
Cauchy–Kovalevskaya theorem
The Cauchy–Kovalevskaya theorem is a fundamental result in partial differential equations that guarantees the existence and uniqueness of analytic solutions to certain initial value problems under appropriate analyticity conditions.
- F. None of above. chosen
Statements (31)
| Predicate | Object |
|---|---|
| instanceOf |
determinant formula
ⓘ
result in linear algebra ⓘ |
| appearsIn |
algebraic combinatorics
ⓘ
classical invariant theory ⓘ |
| appliesTo | Cauchy matrix ⓘ |
| assumes |
x_i + y_j ≠ 0 for all i,j
ⓘ
x_i are pairwise distinct ⓘ y_j are pairwise distinct ⓘ |
| describes | determinant of Cauchy matrix ⓘ |
| era | 19th century mathematics ⓘ |
| field |
linear algebra
ⓘ
matrix theory ⓘ |
| generalizationOf | determinant of Vandermonde-type matrices ⓘ |
| gives | closed-form expression for determinant ⓘ |
| hasClosedForm | product over i<k (x_k - x_i) times product over j<ℓ (y_ℓ - y_j) divided by product over i,j (x_i + y_j) ⓘ |
| matrixEntryForm |
1/(x_i + y_j)
ⓘ
1/(x_i - y_j) ⓘ |
| namedAfter | Augustin-Louis Cauchy ⓘ |
| namedInLanguage |
determinant de Cauchy
ⓘ
linked to:
Cauchy determinant
|
| property |
determinant factors into product of differences of x_i and y_j
ⓘ
determinant is rational function of x_i and y_j ⓘ determinant nonzero if x_i and y_j satisfy distinctness conditions ⓘ |
| relatedTo |
Cauchy matrix
ⓘ
Cauchy–Binet formula ⓘ Vandermonde determinant ⓘ
linked to:
Vandermonde matrix
|
| usedFor | computing determinants of structured matrices ⓘ |
| usedIn |
combinatorics
ⓘ
interpolation theory ⓘ random matrix theory ⓘ representation theory ⓘ theory of special functions ⓘ |
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 determinant Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Cauchy determinant