International
Tables for Crystallography Volume A Spacegroup symmetry Edited by Th. Hahn © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. A, ch. 5.2, p. 86

Symmetry operations are transformations in which the coordinate system, i.e. the basis vectors a, b, c and the origin O, are considered to be at rest, whereas the object is mapped onto itself. This can be visualized as a `motion' of an object in such a way that the object before and after the `motion' cannot be distinguished.
A symmetry operation transforms every point X with the coordinates x, y, z to a symmetrically equivalent point with the coordinates , , . In matrix notation, this transformation is performed byThe matrix W is the rotation part and the column matrix w the translation part of the symmetry operation . The pair (W, w) characterizes the operation uniquely. Matrices W for pointgroup operations are given in Tables 11.2.2.1 and 11.2.2.2 .
Again, we can introduce the augmented matrix (cf. Chapter 8.1 ) The coordinates , , of the point , symmetrically equivalent to X with the coordinates x, y, z, are obtained by or, in short notation, A sequence of symmetry operations can be obtained as a product of matrices .
An affine transformation of the coordinate system transforms the coordinates of the starting point as well as the coordinates of a symmetrically equivalent pointThus, the affine transformation transforms also the symmetryoperation matrix and the new matrix is obtained by
Example
Space group (85) is listed in the spacegroup tables with two origins; origin choice 1 with , origin choice 2 with as point symmetry of the origin. How does the matrix of the symmetry operation 0, 0, z; 0, 0, 0 of origin choice 1 transform to the matrix of symmetry operation , , z; , , 0 of origin choice 2?
In the spacegroup tables, origin choice 1, the transformed coordinates are listed. The translation part is zero, i.e. . In Table 11.2.2.1, the matrix W can be found. Thus, the matrix is obtained:
The transformation to origin choice 2 is accomplished by a shift vector p with components , , 0. Since this is a pure shift, the matrices P and Q are the unit matrix I. Now the shift vector q is derived: . Thus, the matrices and are By matrix multiplication, the new matrix is obtained: If the matrix is applied to x′, y′, z′, the coordinates of the starting point in the new coordinate system, we obtain the transformed coordinates , , , By adding a lattice translation a, the transformed coordinates are obtained as listed in the spacegroup tables for origin choice 2.