Orthogonal matrix
From CGAFaq
A square matrix is called orthogonal if its inverse
equals its transpose
, this means that
Observe that this implies, in particular, that the determinant is ±1.
Orthogonal matrices have a number of useful properties. Columns of a orthogonal matrix treated as vectors make out a orthonormal basis of the space (this means that their are pairwise orthogonal and each one has a unit length). The same of course holds true for rows, as well. Conversely, having an orthonormal basis and putting the vectors side by side, makes an orthogonal matrix.
Two orthonormal bases are said to be compatibly oriented if the (orthogonal) matrix representing the change of variables from one to the other has determinant +1. This happens if and only if one basis can be rotated onto the other.
The linear transformation corresponding to an orthogonal matrix is called a (linear) isometry (or a rigid (linear) transform). If the determinant of the matrix equals +1 then it is a rotation, if it equals −1 it is a reflection. Orthogonal matrices of given dimension form a group, called orthogonal group and denoted
. Orthogonal matrices of determinant 1 (i.e. those associated with rotations) form a subgroup called special orthogonal group and denoted
.
An isometry preserves many geometric properties of the space. In particular it preserves lengths, perpendicularity and angles. Since the inverse and transpose of the matrix of the linear isometry cancel out, normals are treated by an isometry in the same way as ordinary vectors. This is not so for a general transformation (see How are normals transformed?)
