Geometric Algebra for Subspace Operations

TL;DR

This paper introduces a geometric algebra framework for efficient computation of subspace operations like meet and join, applicable across various signatures, enhancing tools for subspace analysis.

Contribution

It develops a unified geometric algebra approach to compute subspace operations, including new methods for join and meet, applicable in any algebra signature.

Findings

Join can be computed even with common factors.

Meet can be computed without knowing the join.

Operations are extendable to any algebra signature.

Abstract

The set theory relations \in, \backslash, \Delta, \cap, and \cup have corollaries in subspace relations. Geometric Algebra is introduced as the ideal framework to explore these subspace operations. The relations \in, \backslash, and \Delta are easily subsumed by Geometric Algebra for Euclidean metrics. A short computation shows that the meet (\cap) and join (\cup) are resolved in a projection operator representation with the aid of one additional product beyond the standard Geometric Algebra products. The result is that the join can be computed even when the subspaces have a common factor, and the meet can be computed without knowing the join. All of the operations can be defined and computed in any signature (including degenerate signatures) by transforming the problem to an analogous problem in a different algebra through a transformation induced by a linear invertible function (a LIFT to a different algebra). The new results, as well as the techniques by which we reach them, add to the tools available for subspace computations.