Functions of Clifford Numbers or Square Matrices

Abstract

It is a simple matter to compute the function of a Clifford number or any square matrix if the function is a polynomial. However difficulties arise for more complicated functions. In the course of dealing with square roots of Clifford numbers, Garret Sobczyk became acquainted with some of the literature [3] and [4] on the generalized spectral decompositions of a linear operator. This decomposition removes these difficulties. Since this approach is not well known, Sobczyk has published a sequence of elegant expository articles [5], [6], and [7] to popularize the application of this method. He has also introduced an improved algorithm in the appendix of [5] to deal with the case for which there are multiple roots in the minimal polynomical for the linear operator. In this paper we will carry this slightly further to obtain an explicit formula for the projection operators.