Binet–Cauchy identity

The Binet–Cauchy identity expresses each maximal minor of a product of rectangular matrices as a sum of products of corresponding minors. For square factors, it reduces to the determinant of a product being the product of the determinants.

Minor (linear algebra): A determinant of a square submatrix, formed by selecting rows and columns from a matrix. The identity pairs minors of the two factors along matching index sets.
Linear map: A function between vector spaces that preserves vector addition and scalar multiplication. Matrices in the identity represent composable linear maps between spaces of possibly different dimensions.
Grassmannian: A space whose points represent fixed-dimensional linear subspaces of a vector space. Its Plücker coordinates are minors that satisfy relations derived from Binet–Cauchy.
Sylvester's determinant identity: An identity relating determinants of products AB and BA when A and B are rectangular matrices of compatible sizes. It relates square determinants of reversed products rather than summing paired minors.
Cauchy–Binet formula: A formula expressing a minor of a matrix product as a sum of products of minors. This is the standard name for the identity's general rectangular-matrix form.
Permutation: A rearrangement of the elements of a finite set. Determinants encode signs of permutations, which govern signs in minor expansions.
Plücker coordinates: Homogeneous coordinates for a subspace, given by the maximal minors of a matrix spanning it. Their transformation under a change of basis follows directly from the minor product formula.
Matrix determinant lemma: A formula for the determinant of a matrix after a low-rank update. It handles a structured perturbation, not the minors of a general matrix product.
Determinant: A scalar associated with a square matrix that encodes invertibility, oriented volume scaling, and other properties. The identity extends determinant multiplicativity to rectangular factors through their minors.
Alternating multilinear form: A multilinear function that vanishes whenever two of its arguments are equal and changes sign when arguments are exchanged. Determinants are alternating multilinear, the property underlying the minor-sum formula.

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Leibniz formula for determinantsRelated: It derives determinant multiplicativity from multilinear expansions related to the permutation formula.