Special Orthogonal Group

Orthogonal Group

The orthogonal group $O(n)$ is the group of all $n\times n$ orthogonal matrices. These are matrices $\mathbf{R}$ such that:

$${\mathbf{R}}^{\top }\mathbf{R}=\mathbf{R}{\mathbf{R}}^{\top }=\mathbf{I}$$

Orthogonal matrices represent linear transformations that preserve the Euclidean inner product and distance (isometries that fix the origin).

Definition (Special Orthogonal Group)

The special orthogonal group $\mathrm{SO}(n)$ is the subgroup of $O(n)$ consisting of matrices with determinant $1$:

$$\mathrm{SO}(n)=\{\mathbf{R}\in O(n)\mid \det (\mathbf{R})=1\}$$

Matrices in $\mathrm{SO}(n)$ represent rotations. Matrices in $O(n)$ with determinant $-1$ represent improper rotations (rotations combined with a reflection).

Properties

• Compactness: $\mathrm{SO}(n)$ and $O(n)$ are compact manifolds.
• Isometry: For any $\mathbf{x},\mathbf{y}\in {\mathbb{R}}^n$ and $\mathbf{R}\in O(n)$, we have $⟨\mathbf{Rx},\mathbf{Ry}⟩=⟨\mathbf{x},\mathbf{y}⟩$.
• Lie Algebra: The Lie algebra of $\mathrm{SO}(n)$, consists of all $n\times n$ skew-symmetric matrices.

2D and 3D Cases

SO(2)

Elements of $\mathrm{SO}(2)$ are rotation matrices in the plane:

$$\mathbf{R}(\theta )=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$$

$\mathrm{SO}(2)$ is an Abelian Group.

SO(3)

Elements of $\mathrm{SO}(3)$ represent rotations in 3D space.

Relation to Physics

In rigid body dynamics, $\mathrm{SO}(3)$ is used to represent the orientation of a rigid body in three-dimensional space.