Algorithmic computation of multivector inverses and characteristic polynomials in non-degenerate Clifford algebras

TL;DR

This paper presents a novel algorithm for computing inverses and characteristic polynomials of multivectors in non-degenerate Clifford algebras, implemented in Maxima, enhancing symbolic and numerical computations.

Contribution

It introduces a variation of the Faddeev-LeVerrier-Souriau algorithm tailored for Clifford algebras, enabling efficient inverse and polynomial calculations.

Findings

Algorithm successfully computes inverses in various Clifford algebras.

Implementation in Maxima demonstrates practical applicability.

Examples validate the algorithm's accuracy and efficiency.

Abstract

The power of Clifford or, geometric, algebra lies in its ability to represent geometric operations in a concise and elegant manner. Clifford algebras provide the natural generalizations of complex, dual numbers and quaternions into non-commutative multivectors. The paper demonstrates an algorithm for the computation of inverses of such numbers in a non-degenerate Clifford algebra of an arbitrary dimension. The algorithm is a variation of the Faddeev-LeVerrier-Souriau algorithm and is implemented in the open-source Computer Algebra System Maxima. Symbolic and numerical examples in different Clifford algebras are presented.