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

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. A1. ch. 1.2, p. 8   | 1 | 2 |

Section 1.2.2.4. Matrix–column pairs and [(n+1)\times(n+1)] matrices

Hans Wondratscheka*

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

1.2.2.4. Matrix–column pairs and [(n+1)\times(n+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\times4)] matrix, but such matrices cannot be multiplied by the usual matrix multiplication and cannot be inverted. However, if one supplements the [(3\times4)] matrix by a fourth row `0 0 0 1', one obtains a [(4\times4)] square matrix which can be combined with the analogous matrices of other mappings and can be inverted. These matrices are called augmented matrices and are designated by open-face letters in this volume:[\specialfonts{\bbsf W} = \left (\matrix{W_{11} & W_{12} & W_{13} & \vphantom{\big(^2}\vrule &w_1 \cr \noalign{\vskip-1.0em}\cr W_{21} & W_{22} & W_{23} & \vphantom{\big(^2}\vrule & w_2 \cr \noalign{\vskip-1.0em}\cr W_{31} & W_{32} & W_{33} & \vphantom{\big(^2_2}\vrule & w_3 \cr \noalign{\vskip-1.0em}\cr \noalign{\vskip.3em} \noalign{\hrule} \cr 0 & 0 & 0 & \vphantom{\big(^2}\vrule & 1 } \right), \,\,\, {\tilde{\bbsf x}} = \left (\matrix{\,\tilde{x}\, \cr \tilde{y} \cr \noalign{\vskip-1.0em}\cr \tilde{z}\cr \noalign{\vskip.3em} \noalign{\hrule} \cr 1} \right), \,\,\, {\bbsf x} = \left (\matrix{\,x\, \cr \noalign{\vskip-1.0em}\cr y \cr \noalign{\vskip-1.0em}\cr z \cr \noalign{\vskip.3em} \noalign{\hrule} \cr 1} \right). \eqno (1.2.2.7)]

In order to write equation (1.2.2.3)[link] as [\specialfonts\tilde{{\bbsf x}}= {\bbsf W}{\bbsf x}] with the augmented matrices [\specialfonts{\bbsf W}], the columns [\tilde{{\bi x}}] and x also have to be extended to the augmented columns [\specialfonts{\bbsf x}] and [\specialfonts\tilde{{\bbsf x}}]. Equations (1.2.2.5)[link] and (1.2.2.6)[link] then become [\specialfonts{\bbsf W}_{\rm 3} = {\bbsf W}_{\rm 2}\,{\bbsf W}_{\rm 1} { \rm\,\, and\,\, } ({\bbsf W})^{\rm -1}=({\bbsf W}^{\rm -1}). \eqno (1.2.2.8)]

The vertical and horizontal lines in the matrix have no mathematical meaning. They are simply a convenience for separating the matrix part from the column part and from the row `0 0 0 1', and could be omitted.

Augmented matrices are very useful when writing down general formulae which then become more transparent and more elegant. However, the matrix–column pair formalism is, in general, advantageous for practical calculations.

For the augmented columns of vector coefficients, see Section 1.2.2.6[link].








































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