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

TL;DR

This paper presents a matrix-based proof of the quaternion representation theorem for four-dimensional rotations, establishing a clear geometric interpretation and a method to derive quaternions from rotation matrices.

Contribution

It introduces an associate matrix construction that characterizes 4D rotation matrices via rank-1 and norm conditions, linking them to quaternion pairs.

Findings

Associate matrix has rank 1 and norm 1 if and only if the matrix is a 4D rotation.

The associate matrix is the dyadic product of two 4D unit vectors representing quaternions.

The proof provides a geometric interpretation through similarity transformations.

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.