Geometric Algebras of Light Cone Projective Graph Geometries

Abstract

A null vector is an algebraic quantity with square equal to zero. I denote the universal algebra generated by taking all sums and products of null vectors over the real or complex numbers by N. The rules of addition and multiplication in N are taken to be the familiar rules of addition and multiplication of real or complex square matrices. A pair of null vectors is positively or negatively correlated if their inner product is positive or negative, respectively. A large class of (Clifford) geometric algebras are isomorphic to real or complex matrix algebras, or a pair of such algebras. I begin the study of the eigenvector-eigenvalue problem of linear operators in the geometric algebra G(1,n) of R^{n+1}, and by restricting to barycentric coordinates, n-simplices whose n+1 vertices are non-zero null vectors. These ideas provide a foundation for a new Cayley-Grassmann Theory of Linear Algebra, with many possible applications in pure-applied areas of science and engineering.