International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 1.2, pp. 13-19

## Section 1.2.2. Matrix description of symmetry operations1

H. Wondratscheka and M. I. Aroyob

aLaboratorium für Applikationen der Synchrotronstrahlung (LAS), Universität Karlsruhe, Germany, and bDepartamento de Física de la Materia Condensada, Universidad del País Vasco (UPV/EHU), Bilbao, Spain

### 1.2.2. Matrix description of symmetry operations1

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#### 1.2.2.1. Matrix–column presentation of isometries

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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 space-group symmetry operations are listed in Tables 1.2.2.1 and 1.2.2.2 as matrices for point-group symmetry operations.

 Table 1.2.2.1| top | pdf | Matrices for point-group symmetry operations and orientation of corresponding geometric elements, referred to a cubic, tetragonal, orthorhombic, monoclinic, triclinic or rhombohedral coordinate system
Symbol of symmetry operation and orientation of geometric elementTransformed coordinates Matrix WSymbol of symmetry operation and orientation of geometric elementTransformed coordinates Matrix WSymbol of symmetry operation and orientation of geometric elementTransformed coordinates Matrix WSymbol of symmetry operation and orientation of geometric elementTransformed coordinates Matrix W
1

 Table 1.2.2.2| top | pdf | Matrices for point-group symmetry operations and orientation of corresponding geometric elements, referred to a hexagonal coordinate system
Symbol of symmetry operation and orientation of geometric elementTransformed coordinates Matrix WSymbol of symmetry operation and orientation of geometric elementTransformed coordinates Matrix WSymbol of symmetry operation and orientation of geometric elementTransformed coordinates Matrix W
1

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).

#### 1.2.2.1.1. Shorthand notation of matrix–column pairs

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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 so-called 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

 (1) The coordinate triplet of stands for the symmetry operation with the rotation partand the translation part . The assignment of the coordinate triplet to the matrix–column pair becomes obvious if one applies the equations (1.2.2.2) for the specific case of (W, w):would beThis symmetry operation is found under space group , No. 96 in the space-group tables of Chapter 2.3 . It is the entry (4) of the first block (the so-called General position block) starting with under the heading Positions. (2) The matrix–column pair is represented in shorthand notation by the coordinate triplet . This is the entry (11) of the general positions of the space group , No. 179 (cf. the space-group tables of Chapter 2.3 ).

#### 1.2.2.2. Combination of mappings and inverse mappings

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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 so-called 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.

#### 1.2.2.3. Matrix–column pairs and (3 + 1) × (3 + 1) matrices

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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 open-face 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 augmented-column 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.

#### 1.2.2.4. The geometric meaning of (W, w)

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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.

• (1) Evaluation of the matrix part W:

• (a) Type of operation: In general the coefficients of the matrix depend on the choice of the basis; a change of basis changes the coefficients, see Section 1.5.2 . However, there are geometric quantities that are independent of the basis.

 (i) The preservation of the handedness of a chiral object, i.e. the question of whether the symmetry operation is a rotation or rotoinversion, is a geometric property which is deduced from the determinant of W: : rotation; : rotoinversion. (ii) The angle of rotation . This does not depend on the coordinate basis. The corresponding invariant of the matrix W is the trace and it is defined by . The rotation angle of the rotation or of the rotation part of a rotoinversion can be calculated from the trace by the formulaThe + sign is used for rotations, the − sign for rotoinversions.

The type of isometry: the types 1, 2, 3, 4, 6 or , , , , can be uniquely specified by the matrix invariants: the determinant and the trace tr():

tr(W)
3210−1−3−2−101
Type 1 6 4 3 2
Order 1 6 4 3 2 2 6 4 6 2

• (b) Rotation or rotoinversion axis: All symmetry operations (except 1 and ) have a characteristic axis (the rotation or rotoinversion axis). The direction u of this axis is invariant under the symmetry operation: The + sign is for rotations, the − sign for rotoinversions.

In the case of a k-fold rotation, the direction u can be calculated by the equationwhere is an arbitrary direction. The direction Y(W)v is invariant under the symmetry operation W as the multiplication with W just permutes the terms of Y. If the application of equation (1.2.2.14) results in , then the direction v is perpendicular to u and another direction v has to be selected. In the case of a roto­inversion W, the direction Y(−W)v gives the direction of the rotoinversion axis. For , .

• (c) Sense of rotation (for rotations or rotoinversions with ): The sense of rotation is determined by the sign of the determinant of the matrix Z, given by = , where u is the vector of equation (1.2.2.14) and x is a non-parallel vector of u, e.g. one of the basis vectors.

Examples are given later.

• (2) Analysis of the translation column w:

 (a) If W is the matrix of a rotation of order k or of a reflection (), then , and one determines the intrinsic translation part (or screw part or glide part) of the symmetry operation, also called the intrinsic translation component of the symmetry operation, by orThe vector with the column of coefficients is called the screw or glide vector. This vector is invariant under the symmetry operation: . Indeed, multiplication with W permutes only the terms on the right side of equation (1.2.2.16). Thus, the screw vector of a screw rotation is parallel to the screw axis. The glide vector of a glide reflection is left invariant for the same reason. It is parallel to the glide plane because (−W + I)(I + W) = O. If t = o holds, then (W, w) describes a rotation or reflection. For , (W, w) describes a screw rotation or glide reflection. One forms the so-called reduced operation by subtracting the intrinsic translation part from (W, w): The column is called the location part (or the location component of the translation part) of the symmetry operation because it determines the position of the rotation or screw–rotation axis or of the reflection or glide–reflection plane in space. (b) The set of fixed points of a symmetry operation is obtained by solving the equation Equation (1.2.2.18) has a unique solution for all roto­inversions (including , excluding ). There is a one-dimensional set of solutions for rotations (the rotation axis) and a two-dimensional set of solutions for reflections (the mirror plane). For translations, screw rotations and glide reflections, there are no solutions: there are no fixed points. However, a solution is found for the reduced operation, i.e. after subtraction of the intrinsic translation part, cf. equation (1.2.2.17)

(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 space-group tables of Part 2 can be found in the corresponding Symmetry operations blocks of the space-group 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 space-group tables of Chapter 2.3 ).

#### Examples

 (1) Consider the symmetry operation [symmetry operation (15) of the General position block of the space group ]. Its matrix–column pair is given by Type of operation: the values of det(W) = 1 and tr(W) = 1 show that the symmetry operation is a fourfold rotation. The direction of rotation axis u: The application of equation (1.2.2.14) with the matrixyields the direction u = [001] of the fourfold rotation axis. Sense of rotation: The negative sense of rotation follows from det(Z) = −1, where the matrix(here, is taken as a vector non-parallel to u). Screw component: The intrinsic translation part (screw component) of the symmetry operation is calculated from Location of the symmetry operation: The location of the fourfold screw rotation is given by the fixed points of the reduced symmetry operation . The set of fixed points is obtained from the equation Following the conventions for the designation of symmetry operations adopted in this volume (cf. Section 1.4.2 and 2.1.3.9 ), the symbol of the symmetry operation , , is given by . (2) The symmetry operation with corresponds to the entry No. (30) of the General position block of the space group , No. 230. Type of operation: the values of det(W) = −1 and tr(W) = 0 show that symmetry operation is a threefold rotoinversion. The direction of rotoinversion axis u:yields the direction from . Sense of rotation: The positive sense of rotation follows from the positive sign of the determinant of the matrix Z, det(Z) = 1, where the matrix(here, is taken as a vector non-parallel to u). Location of the symmetry operation: The solution , of the fixed-point equation of the rotoinversiongives the coordinates of the inversion centre on the rotoinversion axis. An obvious description of a line along the direction and passing through the point is given by the parametric expression . The choice of the free parameter results in the description of the rotoinversion axis found in the Symmetry operation block. The convention adopted in this volume to have zero constant at the z coordinate of the description of the axis determines the specific choice of the free parameter. The geometric characteristics of the symmetry operation are reflected in its symbol . (3) The matrix–column pair (W, w) of the symmetry operation (37) of the General position block of the space group is given by Type of operation: The values of the determinant det(W) = −1 and the trace tr(W) = +1 indicate that the symmetry operation is a reflection. Normal u of the reflection plane: The orientation of the reflection plane in space is determined by its normal u, which is directed along . The direction of u follows from the matrix equationwhere is arbitrary. Glide component: The glide component , determined from the equationindicates that the symmetry operation is a d-glide reflection. As expected, the translation vectorcorresponds to a centring translation. Location of the symmetry operation: The location of the d-glide plane follows from the set of fixed points of the reduced symmetry operationThus, the set of fixed points (the d-glide plane) can be described as . The symbol of the symmetry operation , found in the Symmetry operations block of the space-group table of in Chapter 2.3 , comprises the essential geometric characteristics of the symmetry operation, i.e. its type, glide component and location. It is worth repeating that according to the conventions adopted in the space-group tables of Part 2, the mirror planes are specified by their sets of fixed points and not by the normals to the planes (cf. Section 1.4.2 for more details).

#### 1.2.2.5. Determination of matrix–column pairs of symmetry operations

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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.

 (1) The origin: Let with coordinates be the image of the origin O with coordinates o, i.e. . Examination of the equations (1.2.2.1) shows that = w, i.e. the column w can be determined separately from the coefficients of the matrix W. (2) The coordinate points: We consider the point A. Inserting , in equations (1.2.2.1) one obtains or , . The first column of W is separated from the others, and for the solution only the known coefficients have to be subtracted from the coordinates of the image point of A. Analogously one calculates the coefficients from the image of point B (0, 1, 0) and from the image of point C (0, 0, 1).

#### 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 ?

 (a) Image of the origin O: The origin is left invariant by the reflection part of the mapping; it is shifted by the glide part to 1/2, 1/2, 0 which are the coordinates of . Therefore, . (b) Images of the coordinate points. Neither of the points A and B are affected by the reflection part, but A is then shifted to 3/2, 1/2, 0 and B to 1/2, 3/2, 0. This results in the equations , , for A and , , for B. One obtains , , and . Point : 0, 0, 1 is reflected to and then shifted to . This means , , = or , . (c) The matrix–column pair is thuswhich can be represented by the coordinate triplet [cf. Section 1.2.2.1.1 for the shorthand notation of (W, w)].

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 space-group symmetry operations (General position block) from their symbols (Symmetry operations block) found in the space-group 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 matrix-algebraic development of crystallographic groups. III. Z. Kristallogr. 71, 336–366.