Readers are referred to more detailed representation theory of SO(3) in appropriate
literature [1].
17.4.1 Rotation Axis and Rotation Matrix
In this section we represent a vector such as jxi. We start with showing that any
rotation has a unique presence of a rotation axis. The rotation axis is defined by the
following: Suppose that there is a rigid body with some point within the body fixed.
Here the said rigid body can be that with infinite extent. Then the rigid body exerts
rotation. The rotation axis is a line on which every point is unmoved during the
rotation. As a matter of course, identity matrix E has thee linearly independent
rotation axes. (Practically, this represents no rotation.)
Theorem 17.2 Any rotation matrix R is accompanied by at least one rotation axis.
Unless the rotation matrix is identity, the rotation matrix should be accompanied by
one and only one rotation axis.
Proof As R is an orthogonal matrix of a determinant 1, so are R
T and R
À1 . Then we
have
R À E
ð
Þ
T ¼ R
T
À E ¼ R
À1
À E:
ð17:55Þ
Hence, we get
det R À E
ð
Þ¼det R
T
À E
À
Á ¼ det R
À1
À E
À
Á ¼ det R
À1 E À R
ð
Þ
Â
Ã
¼ det R
À1
À
Á
det E À R
ð
Þ¼det E À R
ð
Þ¼Àdet R À E
ð
Þ:
ð17:56Þ
Note here that with any (3, 3) matrix A of ℝ
3 ,
det A ¼ À À1
ð Þ
9 det A ¼ Àdet ÀA
ð Þ:
This equality holds with ℝ
n (n : odd), but in ℝ
n (n : even) we have det
A ¼ det (ÀA). Then, (17.56) results in a trivial equation 0 ¼ 0 accordingly.
Therefore, the discussion made below only applies to ℝ
n (n : odd). Thus, from
(17.56) we have
det R À E
ð
Þ¼0:
ð17:57Þ
This implies that for
∃
jx 0 i 6 ¼ 0
664
17 Symmetry Groups