Geometric matrix algebra

Abstract

Matrix multiplication was first introduced by Arthur Cayley in 1855 in agreement with the composition of linear transformations. We explore an underlying geometric framework in which matrix multiplication naturally arises from the product of numbers in a geometric (Clifford) algebra. Consequently, all invariants of a linear operator become geometric invariants of the multivectors that they represent. Two different kinds of bases for matrices emerge, a spectral basis of idempotents and nilpotents, and a standard basis of scalars, vectors, bivectors, and higher order k-vectors. The Kronecker product of matrices naturally arises when considering the block structure of a matrix. Conformal geometry of R3 is expressed in terms of the concept of an h-twistor, which is a generalization of a Penrose twistor.