Special Orthogonal Group

A special orthogonal group is the set of matrices that has the following properties:

• They are orthogonal such that $Q^{T}Q=I$
• The determinant is $\det (Q)=+1$.

$$\text{SO}(n)=\left\{\begin{array}{c}\mathbf{R}\in {\mathbb{R}}^{n\times n}|\mathbf{R}{\mathbf{R}}^T=\mathbf{I},\det (\mathbf{R})=1\end{array}\right\}$$

Geometric Interpretation

First, orthogonality preserves dot products, lengths, and angles (see here) between any set of give vectors – for example, two vectors that define a coordinate frame.

Orthogonal matrices always have determinant of $+1$ or $-1$. Matrices with determinant $-1$ reverse the orientation. For example,

$$Q=\left[\begin{array}{cc}1& 0\\ 1& -1\end{array}\right],\det (Q)=-1$$

Applying $Q$ to the standard basis:

$$\begin{array}{rl}{e}_1=(1,0)& ⟶(1,0)\\ e_2=(0,1)& ⟶(0,-1)\end{array}$$

Thus, choosing only matrices where $\det (Q)=+1$ preserves only rotations, not reflections/inversions. This is why rotation matrices for robotics belong to SO(2) or SO(3).