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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
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
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.