The QRD and SVD of matrices over a real algebra

TL;DR

This paper generalizes the QR and SVD decompositions to matrices over a broad class of real algebras, enhancing tools for signal processing applications involving algebraic structures.

Contribution

It introduces two methods for computing QR and SVD in real algebras, extending classical decompositions to semi-simple, group, and Clifford algebras.

Findings

Generalization of QR and SVD to real algebras.

Two computational approaches: Jacobi method and Artin-Wedderburn theorem.

Applicable to a wide class of algebras including semi-simple, group, and Clifford algebras.

Abstract

Recent work in the field of signal processing has shown that the singular value decomposition of a matrix with entries in certain real algebras can be a powerful tool. In this article we show how to generalise the QR decomposition and SVD to a wide class of real algebras, including all finite-dimensional semi-simple algebras, (twisted) group algebras and Clifford algebras. Two approaches are described for computing the QRD/SVD: one Jacobi method with a generalised Givens rotation, and one based on the Artin-Wedderburn theorem.