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

TL;DR

This paper develops noncommutative generalizations of the Cauchy-Binet formula, providing elementary proofs of classical identities like Capelli and Turnbull's identities through simple commutator manipulations.

Contribution

It introduces new noncommutative determinant identities and offers simplified proofs of classical invariant theory results.

Findings

Derived noncommutative Cauchy-Binet formulas

Provided elementary proofs of Capelli identities

Extended identities to symmetric and antisymmetric matrices

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.