Notations

1  Introduction

The classical quaternion representation theorem for rotations in 4D Euclidean space states that an arbitrary 4D rotation matrix is the product of a matrix representing left-multiplication by a unit quaternion and a matrix representing right-multiplication by a unit quaternion. This decomposition is unique up to sign of the pair of component matrices.
In this paper this theorem is proved as a result in the theory of matrices by pure matrix means. As a matrix-theoretic theorem it has no a priori geometrical meaning; for this one has to study the behaviour of its matrix formulation under a predetermined class of similarity transformations.
The proof in this paper is not the first one in the existing literature; presumably BOUMAN ([BOUM 1932]) was the first person to publish a satisfactory proof of the representation theorem.

The application of quaternions to geometry is a well-known classical subject, but in the opinion of the author it is also still a fertile research area. Think only of the use of low-dimensional geometry and topology in certain areas of theoretical physics. Or think of the hidden 4D rotational symmetry of the COULOMB field.
The author's desire to make computer software for 3D and 4D geometry led to the purely matrix- based proof of the quaternion representation theorem for 4D rotations presented in this paper.

2  Theorem and outline of proof

Theorem: Each 4D rotation matrix can be decomposed in two ways into a matrix representing left-multiplication by a unit quaternion and a matrix representing right-multiplication by a unit quaternion. These decompositions differ only in the signs of the component matrices.
 

Outline of proof: Let M_L, M_R be matrices representing left- and right-multiplication by a unit quaternion, respectively. Then their product A = M_LM_R is a 4D rotation matrix.
Matrix M_L is determined by four reals a, b, c, d satisfying the relation a² + b² + c² + d² = 1.
Likewise, matrix M_R is determined by four reals p, q, r, s satisfying the relation p² + q² + r² + s² = 1. The 16 products ap, aq, ar, as, ..., dp, dq, dr, ds are arranged into a matrix M, which has rank 1 and is easily expressed in the elements of A. Let us in this paper denote it as the associate matrix of A. Conversely, given an arbitrary 4D rotation matrix A, one calculates its associate matrix M in the hope that it is a matrix of products ap, aq, ar, as, ..., dp, dq, dr, ds which are not all zero. This hope is vindicated by proving that M has rank 1 whenever A is a 4D rotation matrix.
The proof is completed by observing that the sum of the squares of the elements of M is unity, and concluding that two pairs of quadruples of reals a, b, c, d; p, q, r, s exist satisfying
a² + b² + c² + d² = 1,  p² + q² + r² + s² = 1 and differing only in sign.

3  Quaternions in 4D Euclidean geometry

In this paper the 4D Euclidean space is provided with a Cartesian coordinate system OUXYZ. Points are represented as column vectors (u, x, y, z)^T (in this paper denoted as the R⁴ representation) or as quaternions u + xi + yj + zk. In this connection the coordinate system is also denoted as O1IJK.
Both of these representations are based on an arbitrary choice of coordinate system.
To obtain a possible geometrical meaning of a matrix-algebraic result, one has to prove that it is invariant under coordinate system transformations (similarity transformations), which are in this paper restricted to rotations because we are interested in Euclidean properties, not in more general affine or projective properties.
In the sequel we have to do with geometrical objects as well as their representations with respect to specific coordinate systems. For easy reading object and representation are denoted by the same symbol; the object itself in bold face, its algebraic representations in normal face, with decorations added as needed.

4  Proof of the representation theorem for 4D rotations

4.1  Isoclinic 4D rotations

Let P be an arbitrary 4D point, represented as a quaternion P = u + xi + yj + zk.
Let L = a + bi + cj + dk and R = p + qi + rj + sk be unit quaternions
(a² + b² + c² + d² = 1, p² + q² + r² + s² = 1).
Consider the left- and right-multiplication mappings M_L: P => LP and M_R: P => PR.
In the R⁴ representation M_L and M_R are linear mappings with matrices

One easily proves that both M_L and M_R are orientation-preserving isometries of R⁴ with the origin of coordinates O as a fixed point, i. e. rotations of R⁴ about O.
Both M_L and M_R have the property of rotating all half-lines originating from O through the same angle (arccos a for M_L and arccos p for M_R); such rotations are denoted as isoclinic.
There exists however a subtle difference between M_L and M_R, which is best illustrated by a specific example:

Let M_L and M_R be left- and right-multiplication by the quaternion Q_a= cosa + isina, respectively.
Then M_L acts in both coordinate planes 1I and JK as a rotation through the angle a, while M_R acts in the 1I plane as a rotation through a and in the JK plane as a rotation through -a.

The two kinds of isoclinic rotations are conveniently distinguished as  left- and right-isoclinic.

Conversely, an isoclinic 4D rotation about O different from the non-rotation I and from the central reversion -I is represented by either a left-multiplication or a right-multiplication by a unique unit quaternion and so is either a left- or a right-isoclinic rotation. This theorem is not used in the sequel, but it allows us to speak of "left- or right-isoclinic" instead of the cumbersome "represented by a unit quaternion left- or right-multiplication".
This theorem is presumably due to ROBERT S. BALL; in [BALL 1889] the author does not mention it explicitly as a theorem, but nevertheless gives a proof. However, BALL's proof covers only the case in which the off-diagonal elements of the given matrix are all nonzero. A complete proof is given in [MEBI 1994].

In Section 5 we prove that the left- and right-isocliny properties and the rotation angle are independent of the choice of coordinate system, and therefore correspond to intrinsic geometrical properties.

4.2  General 4D rotations

Let us study the composition of a left-isoclinic and a right-isoclinic rotation.

First apply M_L, then M_R to a 4D point P: one obtains (LP)R in quaternion representation. First apply M_R, then M_L: one obtains L(PR). Quaternion multiplication is associative. Therefore left- and right-isoclinic rotations are commutative, and we have M_LM_RP = M_RM_LP  (R⁴ representation) = LPR (quaternion representation).
In the R⁴ representation the mapping P => LPR has the matrix
 

VAN ELFRINKHOF ([ELFR 1897]) was apparently the first person to treat the relation between quaternion multiplications and 4D rotations in this algebraic way. For this reason a matrix of the form of Eq. 2 with a² + b² + c² + d² = 1 and p² + q² + r² + s² = 1 is in this paper denoted as a Van Elfrinkhof matrix.

A Van Elfrinkhof matrix
 

A =  æ ç ç ç ç ç è a00 a01 a02 a03 a10 a11 a12 a13 a20 a21 a22 a23 a30 a31 a32 a33 ö ÷ ÷ ÷ ÷ ÷ ø

...(3)

is readily interpreted as a set of 16 linear equations in the 16 unknowns ap, aq, ar,  as, ..., dp, dq, dr, ds.
With slightly more work than just an inspection of the plus and minus signs in Eq. 2  one obtains these unknowns, arranged as a matrix, which is in turn written as a dyadic product:
 

M =  æ ç ç ç ç è ap  aq  ar  as  bp  bq  br  bs  cp  cq  cr  cs  dp  dq  dr  ds  ö ÷ ÷ ÷ ÷ ø =  æ ç ç ç ç è a b c d ö ÷ ÷ ÷ ÷ ø æ ç è p  q  r  s ö ÷ ø

...(4)

 

= 1/4 *  æ ç ç ç ç ç è a00+a11+a22+a33 +a10-a01-a32+a23 +a20+a31-a02-a13 +a30-a21+a12-a03 a10-a01+a32-a23 -a00-a11+a22+a33 +a30-a21-a12+a03 -a20-a31-a02-a13 a20-a31-a02+a13 -a30-a21-a12-a03 -a00+a11-a22+a33 +a10+a01-a32-a23 a30+a21-a12-a03 +a20-a31+a02-a13 -a10-a01-a32-a23 -a00+a11+a22-a33 ö ÷ ÷ ÷ ÷ ÷ ø .

...(5)

Matrix M contains at least one nonzero element and consists of mutually proportional columns; therefore it has rank 1.
The sum of the squares of its elements is (a² + b² + c² + d²)(p² + q² + r² + s²) = 1.
In this paper M is denoted as the associate matrix of the matrix A.

4.3  The general rotation as a product of isoclinic rotations

In this paragraph we prove that any 4D rotation matrix can be decomposed into a left- and a right-isoclinic rotation matrix in two ways, differing only in sign. If we can find unit quaternions L = a + bi + cj + dk and R = p + qi + rj + sk such that the given 4D rotation matrix can be written as a Van Elfrinkhof matrix, then we are done.

Let A (Eq. 3) be an arbitrary 4D rotation matrix, and let M, calculated according to Eq. 5, be its associate matrix.
Easy and somewhat tedious calculations show that the sum of the squares of the elements of M is unity, and that all 2-by-2 minors of M are zero. In these calculations one uses the orthogonality of A and the equality of complementary 2-by-2 minors of  4D rotation matrices.
The equality of complementary 2-by-2 minors follows from the general theorem stating that complementary minors of an orthogonal matrix B of arbitrary order are equal or opposite according to Det B being equal to +1 or -1. ([CESA 1904], Art. 71,  p. 54)

It follows that M has rank 1, and there exist reals a,  b,  c,  d;  p,  q,  r,  s satisfying a² + b² + c² + d² = 1,  p² + q² + r² + s² = 1 such that M actually obeys Eq. 4.

This yields two pairs of unit quaternions L = a + bi + cj + dk; R = p + qi + rj + sk, differing only in sign. Finally, Eq. 1 gives the left- and right-isoclinic components, and Eq. 2 the original 4D rotation matrix as a Van Elfrinkhof matrix.

Remark: It is sufficient to prove that the second, third and fourth columns of M are proportional to its first column. So only nine 2-by-2 minors need to be calculated, not all 36 of them. The Appendix shows one of these calculations.

5  Independence of choice of coordinate system

In this section we prove that the properties of left- and right-isocliny and the rotation angle of an isoclinic matrix are invariant under arbitrary rotations of coordinate system. This section is in part a refresher on displacement transformations, similarity transformations, and their interplay.
Let A be a 4D rotation with centre O. Let A be its matrix referred to a Cartesian coordinate system OUXYZ. Let OU'X'Y'Z' a be second Cartesian coordinate system, originating from the first one by a rotation S about O. Rotation A is a displacement transformation; rotation S is a similarity transformation. Let S be the matrix of S referred to OUXYZ and let S_L, S_R be its left- and right-isoclinic components. It does not matter which one of the two possible decompositions is taken. Furthermore, let p be an arbitrary 4D point, p = (u, x, y, z)^T, p' = (u', x', y', z')^T its coordinates referred to OUXYZ, OU'X'Y'Z', respectively. Then p' = S^-1p expresses the new coordinates of p in the old ones, and p = Sp' the old coordinates in the new ones.

Let q = Ap be point p rotated by A. How does similarity transformation S affect rotation matrix A? To find this out, begin at new coordinates p', transform to old coordinates p = Sp', apply rotation matrix A to obtain q = Ap = ASp', and finally transform back to new coordinates, ending at q' = A'p' = S^-1ASp'. We conclude that the matrix A' of A referred to OU'X'Y'Z' equals A' = S^-1AS.

Now we come to prove the intrinsic geometrical nature of the isocliny properties of 4D rotations. Consider the general rotational similarity transformation of the general 4D rotation:
 

where the commutativity of left- and right-isoclinic matrices was applied to sweep all left-isoclinic components to the one side while retaining their order, and the right-isoclinic components to the other side while retaining their order too.
Applying Eq. 6 to a left-isoclinic matrix A_L (set A_R =I) one obtains the transformation formula
 

which shows that A_L' , being the product of left-isoclinic factors, is left-isoclinic. One may attribute the property of left-isocliny to the rotation A_L represented by matrix A_L and all its transforms. This establishes the geometrical nature of this property. In other words, one may speak of left-isoclinic rotations in their own right, irrespective of particular matrix representations.
Likewise, for a right-isoclinic matrix one obtains
 

which shows that A_R' is right-isoclinic. Here too the property of right- isocliny may be attributed to the rotation A_R represented by A_R and all its transforms.
In any rotation matrix of any order the main diagonal elements are the cosines of the angles through which the coordinate axes are rotated. Therefore, in an isoclinic matrix of either kind the main diagonal elements are equal. Their sum, the trace of the matrix, is invariant under similarity transformations. It follows that the rotation angle of an isoclinic matrix is also an invariant. So we may speak of the rotation angle of an isoclinic rotation, irrespective of particular matrix representations.

References

Acknowledgement

Appendix

which equals unity because of the orthogonality of A.

All 2-by-2 minors of M are readily shown to be zero. For instance,
 

the bilinear terms cancel because in the rotation matrix A the complementary minors a₂₂a₃₃ - a₃₂a₂₃ and
a₀₀a₁₁ - a₁₀a₀₁ are equal.

As regards the quadratic terms, we have

therefore the quadratic terms in Eq. 7 cancel too.