Cauchy–Binet formula

E825433

mathematical theorem result in linear algebra

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.

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All labels observed (3)

Label	Occurrences
Cauchy–Binet formula canonical	3
Binet–Cauchy identity	1
Cauchy–Binet theorem	1

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]

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
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 ⓘ linked to: Cauchy–Binet formula
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 ⓘ Jacques Philippe Marie Binet ⓘ
relatedTo	Binet–Cauchy identity ⓘ linked to: Cauchy–Binet formula Jacobi’s formula for determinants ⓘ linked to: Jacobi's theorem on determinants Laplace expansion of determinants ⓘ Sylvester’s determinant identity ⓘ linked to: Sylvester determinant
relatesConcept	determinant ⓘ matrix product ⓘ minors of a matrix ⓘ square submatrices ⓘ subdeterminants ⓘ
requiresCondition	inner dimensions of matrices match ⓘ
usedIn	Grassmann algebra ⓘ 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 ⓘ linked to: Hadamard inequality inequalities for determinants ⓘ properties of rank of matrix products ⓘ

How these facts were elicited

Referenced by (5)

Full triples — surface form annotated when it differs from this entity's canonical label.

Cauchy determinant → relatedTo → Cauchy–Binet formula ⓘ

Hadamard inequality → proofTechnique → Cauchy–Binet formula ⓘ

matrix-tree theorem → proofTechniques → Cauchy–Binet formula ⓘ

Cauchy–Binet formula → relatedTo → Binet–Cauchy identity ⓘ

linked to: Cauchy–Binet formula

Cauchy–Binet formula → hasAlternativeName → Cauchy–Binet theorem ⓘ