Orthogonal group

The Lie group of linear transformations preserving a nondegenerate symmetric bilinear form.

Orthogonal group

Let $(V,\langle\ ,\ \rangle)$ be a finite-dimensional real inner product space.

Definition (Euclidean case)

The orthogonal group is

O(V)=\{A\in \mathrm{GL}(V)\mid \langle Av,Aw\rangle=\langle v,w\rangle\ \text{for all }v,w\in V\}.

After choosing an orthonormal basis of $V\cong \Bbb R^n$ , this becomes

O(n)=\{A\in \mathrm{GL}(n,\Bbb R)\mid A^TA=I\}.

It is a closed subgroup of the general linear group , hence a Lie group by the closed subgroup theorem .

The determinant gives two connected components: $\det=1$ and $\det=-1$ . The identity component is the special orthogonal group $SO(n)$ .

Indefinite signatures

More generally, given a nondegenerate symmetric bilinear form of signature $(p,q)$ , the group of linear transformations preserving it is denoted $O(p,q)$ . The case $O(1,n-1)$ is the Lorentz group .

Lie algebra

The Lie algebra of $O(n)$ is $\mathfrak{so}(n)$ , consisting of skew-symmetric matrices, reflecting the infinitesimal condition for orthogonality.