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A matrix-based proof of the quaternion representation theorem for four-dimensional rotations

Johan Ernest Mebius
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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.

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BibTeXRIS

Johan Ernest Mebius. 2005-01-16. A matrix-based proof of the quaternion representation theorem for four-dimensional rotations. https://arxiv.org/abs/math/0501249

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