A multilinear algebra proof of the Cauchy-Binet formula and a multilinear version of Parseval's identity

TL;DR

This paper presents a multilinear algebra proof of the Cauchy-Binet formula and extends it to Hilbert spaces, leading to a generalized Parseval identity, offering new insights into classical linear algebra and functional analysis.

Contribution

It provides a novel multilinear algebra proof of the Cauchy-Binet formula and generalizes it to Hilbert spaces, resulting in a broader version of Parseval's identity.

Findings

Proof of Cauchy-Binet formula via multilinear algebra

Generalization of the formula to abstract Hilbert spaces

Derivation of a generalized Parseval identity

Abstract

We give a short proof of the Cauchy-Binet determinantal formula using multilinear algebra by first generalizing it to an identity {\em not} involving determinants. By extending the formula to abstract Hilbert spaces we obtain, as a corollary, a generalization of the classical Parseval identity.