The description of mappings

Wondratschek, H.

doi:10.1107/97809553602060000538

International
Tables for
Crystallography
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

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International Tables for Crystallography (2006). Vol. A1. ch. 1.2, pp. 7-8 | 1 | 2 |

Section 1.2.2.3. The description of mappings

Hans Wondratschek^a ^*

^a Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail: wondra@physik.uni-karlsruhe.de

1.2.2.3. The description of mappings

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The instruction for the calculation of the coordinates of $[\tilde{X}]$ from the coordinates of X is simple for an affine mapping and thus for an isometry. The equations are $[\eqalign{\tilde {x} &= W_{11}\,x + W_{12}\,y + W_{13}\,z + w_1 \cr \tilde{y} &= W_{21}\,x + W_{22}\,y + W_{23}\,z + w_2 \cr \tilde{z} &= W_{31}\,x + W_{32}\,y + W_{33}\,z + w_3, }\eqno(1.2.2.1)]$ where the coefficients $[W_{ik}]$ and [w_j] are constant. These equations can be written using the matrix formalism: $[ \left (\matrix{\tilde{x} \cr \tilde{y} \cr \tilde{z}}\right) = \left (\matrix{ W_{11} \,\, W_{12}\, \, W_{13} \cr W_{21}\, \, W_{22}\, \, W_{23} \cr W_{31}\, \, W_{32}\, \, W_{33}} \right) \left (\matrix{ x \cr y \cr z }\right) + \left (\matrix{ w_1 \cr w_2 \cr w_3} \right). \eqno (1.2.2.2)]$ This matrix equation is usually abbreviated by $[{\tilde {\bi x}} = {{\bi W}} \, {{\bi x}} + {{\bi w}}, \eqno (1.2.2.3)]$ where $[\tilde{{\bi x}} = \left (\matrix{\tilde{x} \cr \tilde{y} \cr \tilde{z}} \right)\!, \, \, {\bi x} = \left (\matrix{x \cr y \cr z}\right)\!, \, \, {\bi w} = \left (\matrix{w_1 \cr w_2 \cr w_3 } \right) \, {\rm and }\, \, {\bi W} = \left (\matrix{W_{11} & W_{12} & W_{13} \cr W_{21} & W_{22} & W_{23} \cr W_{31} & W_{32} & W_{33} } \right). ]$

Definition 1.2.2.3.1. The matrix W is called the linear part or matrix part, the column w is the translation part or column part of a mapping.

In equations (1.2.2.1) and (1.2.2.3), the coordinates are mixed with the quantities describing the mapping, designated by the letters $[W_{ik}]$ and [w_j] or W and w. Therefore, one prefers to write equation (1.2.2.3) in the form $[{\tilde {\bi x}} = ({\bi W}, \, {\bi w}) \, {\bi x}\, \, {\rm or } \, \, {\tilde {\bi x}} = \{ {\bi W} \, | \, {\bi w} \} \, {\bi x}. \eqno (1.2.2.4)]$ The symbols ( $[{\bi W}, \, {\bi w}]$ ) and { $[{\bi W} \, | \, {\bi w}]$ } which describe the mapping referred to the chosen coordinate system are called the matrix–column pair and the Seitz symbol.

The formulae for the combination of affine mappings and for the inverse of an affine mapping (regular matrix W) are obtained by $[\eqalign{{\tilde {\bi x}} &= {\bi W}_1 \, {\bi x} + {\bi w}_1, \, {\tilde {\tilde {\bi x}}} = {\bi W}_2 \, {\tilde {\bi x}} + {\bi w}_2 = {\bi W}_3 {\bi x} + {\bi w}_3 \cr {\tilde {\tilde {\bi x}}} &= {\bi W}_2 \, ({\bi W}_1 {\bi x} + {\bi w}_1) + {\bi w}_2 = {\bi W}_2 {\bi W}_1 {\bi x} \, + \, {\bi W}_2{\bi w}_1 + {\bi w}_2. } ]$ From $[{\tilde {\bi x}} = {\bi Wx} + {\bi w}]$ , it follows that $[{\bi W}^{-1} \, {\tilde {\bi x}} = {\bi x} + {\bi W}^{-1} \, {\bi w}]$ or $[{\bi x} = {\bi W}^{-1} \, {\tilde{\bi x}} - {\bi W}^{-1} \, {\bi w}.]$

Using matrix–column pairs, this reads $[({\bi W}_3, \, {\bi w}_3) = ({\bi W}_2, \, {\bi w}_2)\,({\bi W}_1, \, {\bi w}_1)= ({\bi W}_2{\bi W}_1, \, {\bi W}_2 {\bi w}_1 + {\bi w}_2) \eqno (1.2.2.5)]$ and $[{\bi x} = ({\bi W}, \, {\bi w})^{-1} {\tilde{\bi x}} = ({\bi W}', \, {\bi w}') \tilde{ \it x}]$ or $[({\bi W}', \, {\bi w}') = ({\bi W}, \, {\bi w})^{-1} = ({\bi W}^{-1}, \, -{\bi W}^{-1}{\bi w}). \eqno (1.2.2.6)]$

One finds from equations (1.2.2.5) and (1.2.2.6) that the linear parts of the matrix–column pairs transform as one would expect:

(1) the linear part of the product of two matrix–column pairs is the product of the linear parts, i.e. if $[({\bi W}_3, \, {\bi w}_3)=]$ $[({\bi W}_2, \, {\bi w}_2) \, ({\bi W}_1, \, {\bi w}_1)]$ then $[{\bi W}_3 = {\bi W}_2 \, {\bi W}_1]$ ;
(2) the linear part of the inverse of a matrix–column pair is the inverse of the linear part, i.e. if $[({\bi X}, \, {\bi x}) = ({\bi W}, \, {\bi w})^{-1}]$ , then $[{\bi X} = {\bi W}^{-1}]$ . [This relation is included in the first one: from $[({\bi W}, \, {\bi w}) \, ({\bi X}, \, {\bi x})=({\bi W}{\bi X}, {\bi Wx} + {\bi w}) =({\bi I}, \, {\bi o})]$ follows $[{\bi X} = {\bi W}^{-1}]$ . Here I is the unit matrix and o is the column consisting of zeroes].

These relations will be used in Section 1.2.5.4.

For the column parts, equations (1.2.2.5) and (1.2.2.6) are less convenient: $[(1) \,\, {\bi w}_3 = {\bi W}_2 \, {\bi w}_1 + {\bi w}_2; \, \, \qquad (2)\,\, {\bi w}' = - {\bi W}^{-1}{\bi w}.]$

Because of the inconvenience of these relations, it is often preferable to use `augmented' matrices, by which one can describe the combination of affine mappings and the inverse mapping by the equations of the usual matrix multiplication. These matrices are introduced in the next section.

References

International Tables for Crystallography (2006). Vol. A1. ch. 1.2, pp. 7-8