An algorithm for the Cartan-Dieudonn\'e theorem on generalized scalar product spaces

TL;DR

This paper introduces an algorithmic approach using Clifford algebras to decompose orthogonal transformations into reflections in generalized scalar product spaces, extending the Cartan-Dieudonne9 theorem.

Contribution

It provides a novel algorithmic proof for the theorem in spaces with arbitrary signature, linking algebraic structures to geometric transformations.

Findings

Algorithm computes reflection factorizations efficiently.

Extends theorem to spaces with arbitrary signature.

Discusses minimal reflection counts for transformations.

Abstract

We present an algorithmic proof of the Cartan-Dieudonn\'e theorem on generalized real scalar product spaces with arbitrary signature. We use Clifford algebras to compute the factorization of a given orthogonal transformation as a product of reflections with respect to hyperplanes. The relationship with the Cartan-Dieudonn\'e-Scherk theorem is also discussed in relation to the minimum number of reflections required to decompose a given orthogonal transformation.