On the Rigid Rotation Concept in n-Dimensional Spaces

Abstract

A general mathematical formulation of the n × n proper orthogonal matrix, that corresponds to a rigid rotation in n-dimensional real Euclidean space, is given here. It is shown that a rigid rotation depends on an angle (principal angle) and on a set of (n ‒ 2) principal axes. The latter, however, can be more conveniently replaced by only two orthogonal directions that identify the plane of rotation. The inverse problem, that is, how to compute these principal rotation parameters from the rotation matrix, is also treated. In this paper, the Euler Theorem is extended to rotations in n-dimensional spaces by a constructive proof that establishes the relationship between orientation of the displaced orthogonal axes in n dimensions and a minimum sequence of rigid rotations. This fundamental relationship, which introduces a new decomposition for proper orthogonal matrices (those identifying an orientation), can be expressed either by a product or a sum of the same rotation matrices. A similar decomposition in terms of the skew-symmetric matrices is also given. The extension of the rigid rotation formulation to n-dimensional complex Euclidean spaces, is also provided. Finally, we introduce the Ortho-Skew real matrices, which are simultaneously proper orthogonal and skew-symmetric and which exist in even dimensional spaces only, and the Ortho-Skew-Hermitian complex matrices which are orthogonal and Skew-Hermitian. The Ortho-Skew and the Ortho-Skew-Hermitian matrices represent the extension of the scalar imaginary to the matrix field.