Clifford Numbers and their Inverses Calculated using the Matrix Representation

Abstract

The theory of Clifford Algebra includes a statement that each Clifford Algebra is isomorphic to a matrix representation. Several authors discuss this and in particular Ablamowicz [1] gives examples of derivation of the matrix representation. A matrix will itself satisfy the characteristic polynomial equation obeyed by its own eigenvalues. This relationship can be used to calculate the inverse of a matrix from powers of the matrix itself. It is demonstrated that the matrix basis of a Clifford number can be used to calculate the inverse of a Clifford number using the characteristic equation of the matrix and powers of the Clifford number. Examples are given for the algebras Clifford(2), Clifford(3) and Clifford(2,2).