Cauchy–Binet formula
E825433
The Cauchy–Binet formula is a fundamental result in linear algebra that expresses the determinant of a product of two rectangular matrices as a sum of products of determinants of their square submatrices.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Cauchy–Binet formula canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T9844133 — 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–Binet formula Context triple: [Cauchy determinant, relatedTo, Cauchy–Binet formula]
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A.
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.
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B.
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.
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C.
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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D.
Jacobi's theorem on determinants
Jacobi's theorem on determinants is a fundamental result in linear algebra that relates the minors of a matrix to the minors of its adjugate (or inverse), providing key identities used in determinant and matrix theory.
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E.
Cayley–Hamilton theorem
The Cayley–Hamilton theorem is a fundamental result in linear algebra stating that every square matrix satisfies its own characteristic polynomial.
- 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–Binet formula Target entity description: The Cauchy–Binet formula is a fundamental result in linear algebra that expresses the determinant of a product of two rectangular matrices as a sum of products of determinants of their square submatrices.
-
A.
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.
-
B.
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.
-
C.
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.
-
D.
Jacobi's theorem on determinants
Jacobi's theorem on determinants is a fundamental result in linear algebra that relates the minors of a matrix to the minors of its adjugate (or inverse), providing key identities used in determinant and matrix theory.
-
E.
Cayley–Hamilton theorem
The Cayley–Hamilton theorem is a fundamental result in linear algebra stating that every square matrix satisfies its own characteristic polynomial.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in linear algebra ⓘ |
| appearsIn |
advanced linear algebra textbooks
ⓘ
treatises on determinants ⓘ |
| appliesTo |
m×n and n×m matrices with m≤n
ⓘ
rectangular matrices ⓘ |
| consequence | det(AB)=0 if all k×k minors of A or B vanish ⓘ |
| describes | determinant of a product of matrices ⓘ |
| domain | finite-dimensional vector spaces ⓘ |
| expresses | determinant of AB as sum over products of determinants of submatrices of A and B ⓘ |
| field | linear algebra ⓘ |
| generalizationOf | determinant multiplicativity for square matrices ⓘ |
| hasAlternativeName | Cauchy–Binet theorem NERFINISHED ⓘ |
| hasFormulation |
coordinate formulation using minors
ⓘ
exterior algebra formulation using wedge products ⓘ |
| historicalPeriod | 19th-century mathematics ⓘ |
| holdsFor |
matrices over a commutative ring
ⓘ
matrices over a field ⓘ |
| implies | Binet–Cauchy identity for 2×2 determinants as a special case ⓘ |
| involvesOperation |
selection of k columns of first matrix
ⓘ
selection of k rows of second matrix ⓘ summation over all k-element index subsets ⓘ |
| language | symbolic matrix notation ⓘ |
| mathematicalSubjectClassification | 15A15 ⓘ |
| namedAfter |
Augustin-Louis Cauchy
NERFINISHED
ⓘ
Jacques Philippe Marie Binet NERFINISHED ⓘ |
| relatedTo |
Binet–Cauchy identity
NERFINISHED
ⓘ
Jacobi’s formula for determinants NERFINISHED ⓘ Laplace expansion of determinants NERFINISHED ⓘ Sylvester’s determinant identity NERFINISHED ⓘ |
| relatesConcept |
determinant
ⓘ
matrix product ⓘ minors of a matrix ⓘ square submatrices ⓘ subdeterminants ⓘ |
| requiresCondition | inner dimensions of matrices match ⓘ |
| usedIn |
Grassmann algebra
NERFINISHED
ⓘ
combinatorial matrix theory ⓘ exterior algebra ⓘ matrix analysis ⓘ multilinear algebra ⓘ numerical linear algebra ⓘ random matrix theory ⓘ theory of determinants ⓘ |
| usedToProve |
Hadamard’s inequality in some settings
NERFINISHED
ⓘ
inequalities for determinants ⓘ properties of rank of matrix products ⓘ |
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–Binet formula Description of subject: The Cauchy–Binet formula is a fundamental result in linear algebra that expresses the determinant of a product of two rectangular matrices as a sum of products of determinants of their square submatrices.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.