Definition

The subgroup of the general linear group is called the orthogonal group.

The set of matrices with determinant 1 is called the special orthogonal group. (Note that orthogonal matrices all must have determinant .)

Geometrically we can think of the elements of as rotations in -dimensions and are mixtures of rotations and reflections.

Lie group structure

Using the same technique outlined in explanation for why the unitary group is a Lie group, we can see that is a closed subgroup.

Lie algebra

Let be the Lie algebra of and .

Note the Lie algebra of and are the same because all skew-symmetric (real) matrices have trace 0.