20 - The linear products and operations

Abstract

The linear products used in this book are the geometric product, the outer product, the contraction inner products, the scalar product, and the commutator product. They are all linear in their arguments. Examples of unary linear operations that are discussed in this chapter are addition, reversion, grade involution, and grade extraction of multi-vectors. This chapter presents two ways to implement the linear products and operations of geometric algebra. Both implementation approaches are based on the linearity and distributivity of these products and operations. The first approach uses linear algebra to encode the multiplying element as a square matrix acting on the multiplied element, which is encoded as a column matrix. This approach is presented because the matrix ideas are familiar to many people because it is convenient, and because it works for general Clifford algebras. However, it does not exploit the sparseness of most elements in geometric algebra, and is not used much in practice. The second approach is effectively a sparse matrix approach and uses the basis blades idea of the previous chapter, storing multi-vectors as lists of weighted basis blades and literally distributing the work of computing the products and operations to the level of basis blades.