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($4 \times 4$) matrices

The formulae (4.2.6) and (4.2.12) are difficult to keep in mind. It would be fine to have them in a more user-friendly shape. Such a shape exists and will be demonstrated now. It is not only more convenient but also solves another problem, viz the clear distinction between point coordinates and vector coefficients, as will be seen in Section 4.4.

If a crystallographic symmetry operation is described by the matrix-column pair (W,w), then one can form the $(3 \times 4)$ matrix

$\left( \begin{array}{rrrr}
W_{11}&W_{12}&W_{13}&w_1 \\ W_{21}&W_{22}&W_{23}&w_2 \\ W_{31}&W_{32}
&W_{33}&w_3\end{array} \right).$

Regrettably, such matrices can not be multiplied with each other because of the different number (4) of columns of the left matrix and (3) of rows of the right matrix, see Section 2.4. However, one can make the matrix square by adding a fourth row `0 0 0 1'. Such $(4\times 4)$ matrices can be multiplied with each other. For the applications also the coordinate columns have to be extended. This is done by adding a fourth row with the number 1 to the $(3 \times 1)$ column. We thus have:

\begin{displaymath}
\mbox{\textit{\textbf{x}}} \rightarrow \mbox{$\mos{x}$}=\lef...
...x{\textit{\textbf{w}}}\\ &&&\\ \hline0&0&0&1\end{array}\right).\end{displaymath} (4.3.1)

Definition (D 4.3.1) The $(4\times 4)$ matrix $\mbox{$\mos{W}$}$ obtained from W and w in the way just described is called the augmented matrix $\mbox{$\mos{W}$}$; the columns are called augmented columns.

The horizontal and vertical lines in the matrix and the horizontal line in the columns have no mathematical meaning; they are to remind the user of the geometric contents and of the way in which the matrix has been built up.

Equation (4.1.2) is replaced by an equation in outlined letters

\begin{displaymath}\left( \begin{array}{c}\tilde{x}\\ \tilde{y}\\ \tilde{z}\\
\...
... \tilde{\mbox{$\mos{x}$}}=
\mbox{$\mos{W}$}\,\mbox{$\mos{x}$}.
\end{displaymath} (4.3.2)

The augmented matrices may be multiplied, and the product is indeed a $(4\times 4)$ matrix whose matrix and column parts are the same as those obtained from equation (4.2.6):
\begin{displaymath}
\left( \begin{array}{ccc\vert c}&&&\\ &\mbox{\textit{\textbf...
...extit{\textbf{v}}}\\ &&&\\ \hline0&0&0&1
\end{array} \right).
\end{displaymath} (4.3.3)

For the reverse mapping $\mbox{$\mos{W}$}^{-1},\ \ \mbox{$\mos{W}$}^{-1}\mbox{$\mos{W}$}=\mbox{$\mos{I}$}$ holds, where $\mbox{$\mos{I}$}$ is the ($4 \times 4$) unit matrix. This is fulfilled for

\begin{displaymath}
\mbox{$\mos{W}$}^{-1}=\left(
\begin{array}{ccc\vert c} &&&\...
...xtit{\textbf{w}}}\\ &&&\\
\hline0&0&0&1 \end{array} \right),
\end{displaymath} (4.3.4)

which corresponds to equation (4.2.12).

In practice the augmented quantities are very convenient for general formulae and for the actual combination of mappings by multiplying $(4\times 4)$ matrices. Equation (4.3.4) is useful to provide the inverse of a $(4\times 4)$ matrix by calculating the right side. It does not make sense to invert a $(4\times 4)$ matrix using equation (2.6.1) on p. [*] for direct matrix inversion.

In an analogous way one can describe mappings of the plane by $(3\times 3)$ augmented matrices and $(3 \times 1)$ augmented columns.


next up previous contents index
Next: Transformation of vector coefficients Up: The description of mappings by ... Previous: Combination and reversion of mappings

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