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.
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 ⓘ |
Referenced by (2)
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