On SVD and Polar Decomposition in Real and Complexified Clifford Algebras

TL;DR

This paper introduces a novel approach to perform SVD and polar decomposition directly within real and complexified Clifford geometric algebras, avoiding matrix operations and enabling applications across science and engineering.

Contribution

It provides new theorems for SVD and polar decomposition in geometric algebras, defining related structures without matrix reliance, applicable in multiple scientific fields.

Findings

Operations are performed entirely within geometric algebras.

Theorems do not involve matrix computations.

Applicable to various dimensions and signatures.

Abstract

In this paper, we present a natural implementation of singular value decomposition (SVD) and polar decomposition of an arbitrary multivector in nondegenerate real and complexified Clifford geometric algebras of arbitrary dimension and signature. The new theorems involve only operations in geometric algebras and do not involve matrix operations. We naturally define these and other related structures such as Hermitian conjugation, Euclidean space, and Lie groups in geometric algebras. The results can be used in various applications of geometric algebras in computer science, engineering, and physics.