O(n,R) - Knowlpedia

For every integer $n\ge1$ , $\operatorname{O}(n,\mathbb R)$ is the orthogonal group

\operatorname{O}(n,\mathbb R)=\{A\in M_{n}(\mathbb R):A^{\mathsf T}A=I\},

with matrix multiplication. The preserved form is the positive-definite real Euclidean form.

Dimensions and structure

This group has real dimension $n(n-1)/2$ . Its tangent algebra consists of skew-symmetric matrices, $X^{\mathsf T}+X=0$ . It has two connected components distinguished by determinant.

References

Pavel Etingof, Lie Groups and Lie Algebras (MIT lecture notes), §6.1, pp. 38–39; Proposition 6.3 and Exercise 6.5.