Quaternion involutions and anti-involutions

Authors:
Todd A. Ell;Stephen J. Sangwine
Affiliations:
5620 Oak View Court, Savage, MN 55378-4695, USA;Department of Electronic Systems Engineering, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, United Kingdom
Venue:
Computers & Mathematics with Applications
Year:
2007

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Abstract

An involution or anti-involution is a self-inverse linear mapping. In this paper we study quaternion involutions and anti-involutions. We review formal axioms for such involutions and anti-involutions. We present two mappings, one a quaternion involution and one an anti-involution, and a geometric interpretation of each as reflections. We present results on the composition of these mappings and show that the quaternion conjugate may be expressed using three mutually perpendicular anti-involutions. Finally, we show that projection of a vector or quaternion can be expressed concisely using three mutually perpendicular anti-involutions.