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| Cauchy–Binet formula | |
|---|---|
| Name | Cauchy–Binet formula |
| Caption | Determinant expansion for product of matrices |
| Field | Linear algebra |
| Discovered by | Augustin-Louis Cauchy, Pierre Ossian Bonnet |
| Year | 19th century |
Cauchy–Binet formula is a classical identity in linear algebra giving the determinant of the product of a rectangular matrix and its transpose or of two rectangular matrices via sums of minors. It refines the relation between Arthur Cayley's determinant properties, Carl Friedrich Gauss's elimination methods, and the multilinear algebra developed by Hermann Grassmann and Élie Cartan. The formula plays a central role in combinatorial matrix theory, representation theory of the General linear group and applications ranging from Leonhard Euler-style enumerative identities to modern numerical analysis.
Let A be an m×n matrix and B an n×m matrix over a commutative ring; then the determinant det(AB) of the m×m product equals the sum over all k-element subsets of column indices of A and corresponding row indices of B of products of k×k minors. This statement generalizes the determinant multiplicativity proved by Augustin-Louis Cauchy and connects to the expansion techniques used by Pierre-Simon Laplace, James Joseph Sylvester, and Arthur Cayley. In the square case m=n it reduces to det(AB)=det(A)det(B), a relation central to Carl Gustav Jacobi's studies and to the spectral theory developed later by John von Neumann.
Proofs of the Cauchy–Binet formula proceed by multilinearity and combinatorial expansion, by reduction to the case of elementary matrices as used by Isaac Newton-inspired elimination, or by exterior algebra methods attributed to Hermann Grassmann and formalized by Élie Cartan and Alfred North Whitehead. A direct proof expands det(AB) by rows and expresses each entry of AB as a sum over index k, then interchanges summation to collect products of minors, an approach analogous to techniques in Srinivasa Ramanujan's series manipulations and in George Boole's algebraic expansions. Exterior algebra proofs identify det as the action on top exterior power, linking to representations of the General linear group and constructions used in the work of Hermann Weyl and Claude Chevalley.
When m=n the formula yields det(AB)=det(A)det(B), a cornerstone in proofs by Arthur Cayley, Carl Friedrich Gauss, and George Peacock. For m=1 it reproduces classical bilinear forms considered by Carl Jacobi and Adrien-Marie Legendre. The formula implies the Binet formula for Fibonacci numbers through companion matrices connected to Édouard Lucas and underlies identities in combinatorics studied by Paul Erdős and George Pólya. It also yields Sylvester's rank inequality and relations used by Issai Schur in his determinant factorization results and by John von Neumann in numerical stability analyses.
Cauchy–Binet is used in evaluating integrals in random matrix theory developed by Eugene Wigner and Freeman Dyson, in the derivation of formulas in multivariate statistics associated with Karl Pearson and Ronald Fisher, and in control theory originating from Norbert Wiener's work. It appears in proofs in algebraic combinatorics by Richard Stanley and Donald Knuth, in algebraic geometry contexts related to Alexander Grothendieck and Jean-Pierre Serre, and in quantum information where techniques from John Bell and Paul Dirac intersect linear-algebra identities. Numerical linear algebra algorithms by James Hardy Wilkinson exploit the formula for error analysis, and it informs complexity bounds in theoretical computer science influenced by Alan Turing and Leslie Valiant.
Generalizations include the matrix determinant lemma related to work by Edmond Landau and the Jacobi–Trudi identities from Giuseppe Peano-inspired symmetric function theory; multilinear extensions employ exterior algebra and Schur functors studied by Isaac Schur and Alfred Young. Noncommutative analogs have been explored in the theory of quantum groups connected to Drinfeld and Vladimir Drinfeld-style deformations and in the context of operator algebras influenced by John von Neumann and Alain Connes. Further extensions relate to the MacMahon master theorem investigated by Percy MacMahon and to identities in tropical geometry appearing in modern work by Grigory Mikhalkin.
For 2×3 matrix A and 3×2 matrix B the determinant det(AB) equals the sum of products of 2×2 minors corresponding to the three choices of column pairs, a calculation akin to expansions used by Pierre-Simon Laplace and examples found in texts by Augustin-Louis Cauchy. For square 2×2 matrices the formula reproduces det(AB)=det(A)det(B), a fact central to classical computations by Carl Friedrich Gauss and Adrien-Marie Legendre. In combinatorial applications, the determinant of a product of incidence-like matrices yields enumerative counts studied by George Pólya and Harold Davenport.