Chapter 20 - The Linear Products and Operations

Abstract

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 the 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. The second approach is effectively a sparse matrix approach storing multivectors as lists of weighted basis blades and distributing the work of computing the products and operations to the level of basis blades. A multivector from an n-dimensional geometric algebra can be stored as a 2n × 1 column matrix. Each element in the matrix is a coordinate that refers to a specific basis blade. The elements of geometric algebra form a linear space, and these linear operations are implemented trivially in the matrix approach that involves the addition of elements performed by adding the matrices. The unary linear operations of reversion, grade involution, and Clifford conjugation can also be implemented as matrices. The entries of these matrices need to be set according to the corresponding operations on the basis blades in row matrix L.