Special Orthogonal Group
Definition
Denote by $\operatorname{SO}(n)$ the set of $n \times n$ orthogonal matrices whose determinant is $+1$, and call it the $n$-dimensional special orthogonal group.
$$ \operatorname{SO}(n) := {\left\{ A \in M_{n \times n}(\mathbb{R}) : AA^{T} = I \quad\text{and}\quad \det(A) = +1 \right\}} $$
Explanation
Basically, orthogonal matrices have determinant either $+1$ or $-1$. Among these, those with determinant $+1$ are rotation matrices, so the special orthogonal group is the set of rotation matrices. Hence it is also called the rotation group. In low dimensions the special orthogonal groups are as follows.
- $\operatorname{SO}(1) \cong \left\{ 1 \right\}$
- $\operatorname{SO}(2) \cong$ $S^{1}$
- $\operatorname{SO}(3) \cong \mathbb{P}^{3}$
Note that $\operatorname{SO}(2)$ is $1$-dimensional, whereas $\operatorname{SO}(3)$ is $3$-dimensional. Rotations in the plane can be expressed with $1$ parameters, while $3$-dimensional space is formed from three planes (the $xy–$ plane, the $yz–$ plane, and the $zx–$ plane), so $3$ parameters are required.
Two dimensions
The 2-dimensional rotation group $\operatorname{SO}(2)$ is concretely given by:
$$ \operatorname{SO}(2) = \left\{ \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} : \theta \in \mathbb{R} \right\} $$
This is isomorphic to the points on the unit circle.
$$ \operatorname{SO}(2) \cong S^{1} $$