CLIFFORD GEOMETRIC ALGEBRAS IN MULTILINEAR ALGEBRA AND NON-EUCLIDEAN GEOMETRIES

Abstract Given a quadratic form on a vector space, the geometric algebra of the corresponding pseudo-euclidean space is dened in terms of a sim-ple set of rules which characterizes the geometric product of vectors. We develop geometric algebra in such a way that it augments, but re-mains fully compatible with, the more traditional tools of matrix alge-bra. Indeed, matrix multiplication arises naturally from the geometric multiplication of vectors by introducing a spectral basis of mutually annihiliating idempotents in the geometric algebra. With the help of a few more algebraic identities, and given the proper geometric inter-pretation, the geometric algebra can be applied to the study of ane, projective, conformal and other geometries. The advantage of geometric algebra is that it provides a single algebraic framework with a compre-hensive, but exible, geometric interpretation. For example, the ane plane of rays is obtained from the euclidean plane of points by adding a single anti-commuting vector to the underlying vector space. The key to the study of noneuclidean geometries is the denition of the oper-ations of meet and join, in terms of which incidence relationships are expressed. The horosphere provides a homogeneous model of euclidean space, and is obtained by adding a second anti-commuting vector to the underlying vector space of the ane plane. Linear orthogonal trans-formations on the higher dimensional vector space correspond to con-formal or Mobius transformations on the horosphere. The horosphere was rst constructed by F.A. Wachter (1792{1817), but has only re-cently attracted attention by oering a host of new computational tools I gratefully acknowledge the support given by INIP of the Universidad de las Americas. The author is a member of SNI, Exp. 14587. 2in projective and hyperbolic geometries when formulated in terms of geometric algebra