Principal angles and approximation for quaternionic projections

TL;DR

This paper extends the concept of principal angles to quaternionic spaces, providing a method to analyze and optimize the approximation of nearly commuting projections by commuting ones using algebraic techniques.

Contribution

It introduces a quaternionic extension of principal angles and develops an algorithm to approximate almost commuting projections with commuting projections.

Findings

Extended Jordan's principal angles to quaternionic spaces

Developed an algorithm for approximating almost commuting projections

Proved properties using universal real C*-algebra

Abstract

We extend Jordan's notion of principal angles to work for two subspaces of quaternionic space, and so have a method to analyze two orthogonal projections in M_n(A) for A the real, complex or quaternionic field (or skew field). From this we derive an algorithm to turn almost commuting projections into commuting projections that minimizes the sum of the displacements of the two projections. We quickly prove what we need using the universal real C*-algebra generated by two projections.