International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 
International Tables for Crystallography (2016). Vol. A, ch. 1.2, pp. 1319
Section 1.2.2. Matrix description of symmetry operations^{1} 
1.2.2. Matrix description of symmetry operations^{1}
In order to describe mappings analytically one introduces a coordinate system , consisting of three linearly independent (i.e. not coplanar) basis vectors (or ) and an origin O. Referred to this coordinate system each point P can be described by three coordinates (or ). A mapping can be regarded as an instruction for how to calculate the coordinates of the image point from the coordinates of the original point .
The instruction for the calculation of the coordinates of from the coordinates of X is simple for an affine mapping and thus for an isometry. The equations arewhere the coefficients and are constant. These equations can be written using the matrix formalism:This matrix equation is usually abbreviated by whereThe matrix W is called the linear part or matrix part and the column w is the translation part or column part of the mapping. The rotation parts W referring to conventional coordinate systems of all spacegroup symmetry operations are listed in Tables 1.2.2.1 and 1.2.2.2 as matrices for pointgroup symmetry operations.


Very often, equation (1.2.2.3) is written in the form The symbols and which describe the mapping referred to the chosen coordinate system are called the matrix–column pair and can be considered as Seitz symbols (Seitz, 1935) (cf. Section 1.4.2.2 for an introduction to and listings of Seitz symbols of crystallographic symmetry operations).
In crystallography in general, and in this volume in particular, an efficient procedure is used to condense the description of symmetry operations by matrix–column pairs considerably. The socalled shorthand notation of the matrix–column pair (W, w) consists of a coordinate triplet , , . All coefficients `+1' and the terms with coefficients 0 are omitted, while coefficients `−1' are replaced by `−' and are frequently written on top of the variable: instead of etc. The following examples illustrate the assignments of the coordinate triplets to the matrix–column pairs.
Examples
The combination of two symmetry operations and is again a symmetry operation. The linear and translation part of the combined symmetry operation is derived from the rotation and translation parts of and in a straightforward way:
Applying first the symmetry operation , on the one hand, On the other hand By comparing equations (1.2.2.5) and (1.2.2.6) one obtainsThe formula for the inverse of an affine mapping follows from the equations , i.e. , which compared with gives Because of the inconvenience of these relations, especially for the column parts of the isometries, it is often preferable to use socalled augmented matrices, by which one can describe the combination of affine mappings and the inverse mapping by equations of matrix multiplication. These matrices are introduced in the following section.
It is natural to combine the matrix part and the column part describing an affine mapping to form a (3 × 4) matrix, but such matrices cannot be multiplied by the usual matrix multiplication and cannot be inverted. However, if one supplements the (3 × 4) matrix by a fourth row `0 0 0 1', one obtains a (4 × 4) square matrix which can be combined with the analogous matrices of other mappings and can be inverted. These matrices are called augmented matrices, and here they are designated by openface letters. Similarly, the columns and x also have to be extended to the augmented columns and :The horizontal and vertical lines in the augmented matrices are useful to facilitate recognition of their coefficients; they have no mathematical meaning.
Equations (1.2.2.1), (1.2.2.7) and (1.2.2.8) then become In the usual description by columns, the vector coefficients cannot be distinguished from the point coordinates, but in the augmentedcolumn description the difference becomes visible.
If and are the augmented columns of coordinates of the points and , and , then is the augmented column of the coefficients of the vector v between P and Q. The last coefficient of is zero, because 1 − 1 = 0. Thus, the column of the coefficients of a vector is not augmented by `1' but by `0'. From the equation for the transformation of the vector coefficients it becomes clear that when the point P is mapped onto the point by according to equation (1.2.2.3), then the vector is mapped onto the vector by transforming its coefficients by . This is because the coefficients are multiplied by the number `0' augmenting the column . Indeed, the vector is not changed when the whole space is mapped onto itself by a translation.
Given the matrix–column pair (W, w) of a symmetry operation , the geometric interpretation of , i.e. the type of operation, screw or glide component, location etc., can be calculated provided the coordinate system to which (W, w) refers is known.
(Note that the reduced operation of a translation is the identity, whose set of fixed points is the whole space.)
The formulae of this section enable the user to find the geometric contents of any symmetry operation. In practice, the geometric meanings for all symmetry operations which are listed in the General position blocks of the spacegroup tables of Part 2 can be found in the corresponding Symmetry operations blocks of the spacegroup tables. The explanation of the symbols for the symmetry operations is found in Sections 1.4.2 and 2.1.3.9 .
The procedure for the geometric interpretation of the matrix–column pairs (W, w) of the symmetry operations is illustrated by three examples of the space group , No. 230 (cf. the spacegroup tables of Chapter 2.3 ).
Examples

The specification of the symmetry operations by their types, screw or glide components and locations is sufficient to determine the corresponding matrix–column pairs (W, w). The general idea is to determine the image points of some points X under the symmetry operation by applying geometrical considerations. The 12 unknown coefficients of (W, w) (nine coefficients and three coefficients ) can then be calculated as solutions of 12 inhomogeneous linear equations obtained from the system of equations (1.2.2.1) written for four pairs (point image point), provided the points X are linearly independent. In fact, because of the special form of the matrix–column pairs, in many cases it is possible to reduce and simplify considerably the calculations necessary for the determination of (W, w): the determination of the image points of the origin O and of the three `coordinate points' , and under the symmetry operation is sufficient for the determination of its matrix–column pair.

Example
What is the pair (W, w) for a glide reflection with the plane through the origin, the normal of the glide plane parallel to c, and with the glide vector ?

The problem of the determination of (W, w) discussed above is simplified if it is reduced to the special case of the derivation of matrix–column pairs of spacegroup symmetry operations (General position block) from their symbols (Symmetry operations block) found in the spacegroup tables of Part 2 of this volume. The main simplification comes from the fact that for all symmetry operations of space groups, the rotation parts W referring to conventional coordinate systems are known and listed in Tables 1.2.2.1 and 1.2.2.2. In this way, given the symbol of the symmetry operation and using the tabulated data, one can write down directly the corresponding rotation part W.
The translation part w of the symmetry operation has two components: . The intrinsic translation part (or screw or glide component) is given explicitly in the symmetry operation symbol. The location part of w is derived from the equationsHere, are the coordinates of an arbitrary fixed point of the symmetry operation.
Example
Consider the symbol of the symmetry operation No. (11) of the Symmetry operations block of the space group (230). The corresponding rotational part W is read directly from Table 1.2.2.1:The location part is determined by the matrix equations[cf. Equation (1.2.2.20)]. The point with coordinates , , is on the screw axis of , i.e. one of the fixed points of the reduced symmetry operation . The translation part w of the matrix–column pair of the symmetry operation is given by The coordinate triplet , corresponding to the derived matrix–column paircoincides exactly with the coordinate triplet listed under No. (11) in the block of the General positions of the space group .
References
Seitz, F. (1935). A matrixalgebraic development of crystallographic groups. III. Z. Kristallogr. 71, 336–366.