Special Orthogonal Group (Rotation)

Overview

$SO(n)$ the special orthogonal group is:

• the orientation preserving isometries of ${\mathbb{R}}^n$ that fix $O$.
• linear transformations that preserves length and orientation
• rotations in ${\mathbb{R}}^n$
• a normal subgroup of Orthogonal Group

Recall from linear algebra, linear transformations that preserves length and orientation have the determinant $1$

Rotation criterion

NOTE

An $n\times n$ real matrix $A$ represents a rotation of ${\mathbb{R}}^n$ if and only if $A{A}^{\top }=I$ and $\det (A)=1$

($A{A}^{\top }=I⟺A^{\top }=A^{-1}⟺A^{\top }A=I$) Proof: $⟸$ using definition of orthogonal matrix, $A$ is a orthogonal matrix. Using Properties we get $\det (A)=\pm 1$ According to properites of orthogonal matrix $A$ preserve norm and inner product of vectors. Since $A$ is square matrix, the converse statement is also true: $A$ is orthogonal $⟺$ $A$ preserves inner product of vectors $⟺$ $A$ preserves norm. If we restrict $\det (A)=1$ then $A$ must be orientation preserving and thus a rotation.

For proof $⟹$, note that matrices that preserve length but not necessarily orientations ($\det (A)=\pm 1$), are called orthogonal (see Orthogonal Group), and if $\det (A)=1$, the matrix is called special orthogonal

Warning

Orthogonal matrices are not the only matrices that preserve orthogonality. A counterexample is dilation matrix $kI$ for $k\ne 0$ Note that preserving inner product is stronger than preserving orthogonality, but is equally strong as preserving length(norm):

Preserving inner product means preserving lengths, angles, and orthogonality

Example

• SO(3): rotation in ${\mathbb{R}}^3$
• SO(4): rotation in ${\mathbb{R}}^4$

Path-Connectedness

We can prove the path-connectedness of $SO(n)$ by induction from $SO(2)$ (obvious path-connected)