Generalized Projection Operators in Geometric Algebra

Abstract

Given an automorphism and an anti-automorphism of a semigroup of a Geometric Algebra, then for each element of the semigroup a (generalized) projection operator exists that is defined on the entire Geometric Algebra. A single fundamental theorem holds for all (generalized) projection operators. This theorem makes previous projection operator formulas equivalent to each other. The class of generalized projection operators includes the familiar subspace projection operation by implementing the automorphism `grade involution' and the anti-automorphism `inverse' on the semigroup of invertible versors. This class of projection operators is studied in some detail as the natural generalization of the subspace projection operators. Other generalized projection operators include projections onto any invertible element, or a weighted projection onto any element. This last projection operator even implies a possible projection operator for the zero element.