Matrix exponential via Clifford algebras

TL;DR

This paper introduces a novel method for computing matrix exponentials by leveraging isomorphisms between matrix algebras and Clifford algebras, enabling symbolic and efficient calculations.

Contribution

It presents a new approach using Clifford algebra isomorphisms and a Maple package to compute matrix exponentials for real, complex, and quaternionic matrices.

Findings

Successfully computes matrix exponentials using Clifford algebra methods.

Demonstrates the approach with three example matrices.

Provides a Maple package for practical implementation.

Abstract

We use isomorphism $\varphi$ between matrix algebras and simple orthogonal Clifford algebras $\cl(Q)$ to compute matrix exponential ${e}^{A}$ of a real, complex, and quaternionic matrix A. The isomorphic image $p=\varphi(A)$ in $\cl(Q),$ where the quadratic form $Q$ has a suitable signature $(p,q),$ is exponentiated modulo a minimal polynomial of $p$ using Clifford exponential. Elements of $\cl(Q)$ are treated as symbolic multivariate polynomials in Grassmann monomials. Computations in $\cl(Q)$ are performed with a Maple package `CLIFFORD'. Three examples of matrix exponentiation are given.