Editing Orthogonal matrix (section)

===Higher dimensions===
Regardless of the dimension, it is always possible to classify orthogonal matrices as purely rotational or not, but for {{nowrap|3 × 3}} matrices and larger the non-rotational matrices can be more complicated than reflections. For example,
<math display="block">
\begin{bmatrix}
-1 & 0 & 0\\
0 & -1 & 0\\
0 & 0 & -1
\end{bmatrix}\text{ and }
\begin{bmatrix}
0 & -1 & 0\\
1 & 0 & 0\\
0 & 0 & -1
\end{bmatrix}</math>

represent an ''[[Inversion in a point|inversion]]'' through the origin and a ''[[improper rotation|rotoinversion]]'', respectively, about the {{math|z}}-axis.

Rotations become more complicated in higher dimensions; they can no longer be completely characterized by one angle, and may affect more than one planar subspace. It is common to describe a {{nowrap|3 × 3}} rotation matrix in terms of an [[axis and angle]], but this only works in three dimensions. Above three dimensions two or more angles are needed, each associated with a [[plane of rotation]].

However, we have elementary building blocks for permutations, reflections, and rotations that apply in general.