A matrix-based proof of the quaternion representation theorem for four-dimensional rotations
Abstract
To each 4x4 matrix of reals another 4x4 matrix is constructed, the so-called associate matrix. This associate matrix is shown to have rank 1 and norm 1 (considered as a 16D vector) if and only if the original matrix is a 4D rotation matrix. This rank-1 matrix is the dyadic product of a pair of 4D unit vectors, which are determined as a pair up to their signs. The leftmost factor (the column vector) consists of the components of the left quaternion and represents the left-isoclinic part of the 4D rotation. The rightmost factor (the row vector) likewise represents the right quaternion and the right-isoclinic part of the 4D rotation. Finally the intrinsic geometrical meaning of this matrix-based proof is established by means of the usual similarity transformations.
Keywords
Cite
@article{arxiv.math/0501249,
title = {A matrix-based proof of the quaternion representation theorem for four-dimensional rotations},
author = {Johan Ernest Mebius},
journal= {arXiv preprint arXiv:math/0501249},
year = {2007}
}
Comments
5 references; about 3000 words