标题： 几何代数中的广义投影算子 显示英文标题

标题： Generalized Projection Operators in Geometric Algebra

摘要： 给定一个几何代数的半群的一个自同构和一个反自同构，那么对于该半群的每个元素，都存在一个(广义)投影算子，该算子定义在整个几何代数上。 所有(广义)投影算子都遵循一个基本定理。 这个定理使先前的投影算子公式彼此等价。 通过在可逆旋量的半群上实现自同构“阶逆转”和反自同构“逆”，广义投影算子类包括了熟悉的子空间投影操作。 这类投影算子被详细研究，作为子空间投影算子的自然推广。 其他广义投影算子包括对任何可逆元素的投影，或者对任何元素的加权投影。 最后一个投影算子甚至暗示了零元素的可能投影算子。 显示英文摘要

摘要： 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.