Hybrid Matrix Geometric Algebra

Abstract

AbstractThe structures of matrix algebra and geometric algebra are completely compatible and in many ways complimentary, each having their own advantages and disadvantages. We present a detailed study of the hybrid 2 × 2 matrix geometric algebra M(2,IG) with elements in the 8 dimensional geometric algebra IG=IG 3 of Euclidean space. The resulting hybrid structure, isomorphic to the geometric algebra IG 4,1 of de Sitter space, combines the simplicity of 2× 2 matrices and the clear geometric interpretation of the elements of IG. It is well known that the geometric algebra IG(4,1) contains the 3-dimensional affine, projective, and conformal spaces of Möbius transformations, together with the 3-dimensional horosphere which has attracted the attention of computer scientists and engineers as well as mathematicians and physicists. In the last section, we describe a sophisticated computer software package, based on Wolfram’s Mathematica, designed specifically to facilitate computations in the hybrid algebra.KeywordsReal VectorGeneral ElementComplex VectorMatrix AlgebraGeometric AlgebraThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.