# A Symbolic Algorithm for Computation of Non-degenerate Clifford Algebra Matrix Representations

## TL;DR

This paper presents an algorithmic method to construct matrix representations of non-degenerate Clifford algebras, facilitating automated proof checking and inverse computation of multivectors.

## Contribution

It introduces a transparent, algorithmic approach for representing Clifford algebras as matrices, enabling automated calculations and proof verification.

## Key findings

- Provides a systematic construction of matrix representations for Clifford algebras.
- Develops an algorithm for computing multivector inverses using matrix representations.
- Demonstrates the application of the Faddeev-LeVerrier-Souriau algorithm in this context.

## Abstract

Modern advances in general-purpose computer algebra systems offer solutions to a variety of problems, which in the past required substantial time investments by trained mathematicians. An excellent example of such development are the Clifford algebras. The main objective of the paper is to demonstrate an utterly algorithmic construction of a Clifford algebra matrix algebra representation of a non-degenerate signature (p, q). While this is not the most economical way of implementation, it offers a transparent mechanism of translation between a Clifford algebra and its faithful real-valued matrix representation and can be used for automated proof checking. This representation is used to derive an algorithm for the computation of an arbitrary multivector inverse as a proof certificate. The proposed algorithm is a mapping of the Faddeev--LeVerrier--Souriau algorithm for computation of the characteristic polynomial of matrices.