Cauchy–Binet formula

The Cauchy–Binet formula expresses the determinant of a product of rectangular matrices as a sum of products of corresponding maximal minors.

Minor (linear algebra): A minor is the determinant of a square submatrix of a matrix. Each summand pairs minors chosen from the two matrices.
Gram determinant: A Gram determinant is the determinant of a matrix of pairwise inner products of vectors. Cauchy–Binet expresses it as a sum of squared maximal minors.
Rectangular matrix: A rectangular matrix has different numbers of rows and columns. The formula's distinctive case uses compatible factors that are not necessarily square.
Augustin-Louis Cauchy: Augustin-Louis Cauchy was a French mathematician whose work helped establish rigorous analysis and modern determinant theory. His name is attached to the identity and its early determinant formulations.
Exterior algebra: Exterior algebra is an algebra of alternating products of vectors and linear transformations. The induced maps on exterior powers turn the identity into a matrix-product rule.
Gram matrix: A Gram matrix records pairwise inner products among a collection of vectors. Applying the formula to a matrix and its transpose yields its Gram determinant.
Submatrix: A submatrix is formed by selecting specified rows and columns of a matrix. The formula selects matching sets of intermediate indices to form its minors.
Jacques Philippe Marie Binet: Jacques Philippe Marie Binet was a French mathematician known for work in mechanics, astronomy, and algebra. His name shares the formula's eponym, reflecting its nineteenth-century origins.
Determinant: A determinant is a scalar associated with a square matrix that encodes volume scaling and invertibility. The formula derives a determinant of a product from determinants of square submatrices.
Cauchy–Binet identity for determinants: A determinant identity relating a product of square matrices to the product of their determinants. For square factors, multiplicativity replaces the rectangular sum over minors.

Linked from 2 pages