Noncommutative determinants, Cauchy–Binet formulae, and Capelli-type identities I. Generalizations of the Capelli and Turnbull identities

Abstract

We prove, by simple manipulation of commutators, two noncommutative generalizations of the Cauchy–Binet formula for the determinant of a product. As special cases we obtain elementary proofs of the Capelli identity from classical invariant theory and of Turnbull's Capelli-type identities for symmetric and antisymmetric matrices.