Rest of Article in dvi style  Matrix exponential via Clifford  Matrix exponential via Clifford

Abstract

Computation of a matrix exponential e^At is needed when solving a system of differential equations or when studying a structure of a linear operator. Standard linear algebra methods depend on solving the eigenvalue problem for A and then on whether A has a complete set of eigenvectors. While a formal power series definition of e^A allows one to represent solutions to such systems in the form $e^{at}{\bf x}_0$ even when A is not diagonalizable, generally there is no good way to implement this definition on a computer. In this paper we describe another way to exponentiate a matrix, let it be numeric or symbolic, with real, complex, or quaternionic entries, totally different from the linear algebra methods. It relies on a well-known isomorphism between matrix algebras and Clifford algebras with the crucial exponentiation done inside an appropriate orthogonal Clifford algebra. This is not a matrix method in the sense that elements of the Clifford algebra are viewed here as multi-variate polynomials in some basis Grassmann monomials with real coefficients. Given a square matrix A over the reals, complexes, or quaternions, one finds its isomorphic image in a suitable Clifford algebra $c \kern -0.1em \ell(q)$ in a form of a polynomial p, one then exponentiates p using Clifford exponentiation, and then one finds a pre-image of e^p which is the desired e^A. This approach requires that one be able to compute spinor representations of orthogonal Clifford algebras which can be easily accomplished with a Maple package 'CLIFFORD'.

Keywords: Clifford algebra, primitive idempotent, spinor representation, Grassmann algebra, bivector, quaternions, matrix norm, polynomial norm, spinor norm.