0. Orthogonal matrices in general

Definition. An $n \times n$ real matrix $P$ is orthogonal if its rows are an orthonormal set of vectors: mutually perpendicular and of length 1. A homogeneous linear transformation is said to be orthogonal if its matrix is.

Examples are the identity matrix $I$ and rotations and reflections (discussed below), for any $n$.

Corollary 0.1 . If $P$ is orthogonal, then $\det P = \pm 1$.

Corollary 0.2 . (a) The product of two orthogonal matrices is orthogonal. (b) The inverse of an orthogonal matrices is orthogonal.

Remark. The standard name ``orthogonal matrix'' is unfortunate; ``orthonormal matrix'' would have been better.