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Subsections


Applications

Only a few applications can be dealt with here:

  1. The inversion of a matrix;
  2. Another formula for calculating the distance between points or the angle between lines (bindings);
  3. A formula for the volume of the unit cell of a crystal.

Inversion of a matrix

The inversion of a square matrix A is a task which occurs everywhere in matrix calculations. Here we restrict the considerations to the inversion of (2 $\times$ 2) and (3 $\times$ 3) matrices. In Least-Squares refiments the inversion of huge matrices was a serious problem before the computers and programs were sufficiently developed.

Definition (D 2.6.2) A matrix C which fulfills the condition $\mbox{\textit{\textbf{C}}}\,\mbox{\textit{\textbf{A}}}=\mbox{\textit{\textbf{I}}}$ for a given matrix A, is called the inverse matrix or the inverse A$^{-1}$ of A.

The matrix A$^{-1}$ exists if and only if $\det{(\mbox{\textit{\textbf{A}}})}\ne 0$. In the following we assume C to exist. If $\mbox{\textit{\textbf{C}}}\,\mbox{\textit{\textbf{A}}}=\mbox{\textit{\textbf{I}}}$ then also $\mbox{\textit{\textbf{A}}}\,\mbox{\textit{\textbf{C}}}=\mbox{\textit{\textbf{I}}}$ holds, i.e. there is exactly one inverse matrix of A. There are two possibilities to calculate the inverse matrix of a given matrix. The first one is particularly simple but not always applicable. The other may be rather tedious but always works.

Definition (D 2.6.2) A matrix A is called orthogonal if $\mbox{\textit{\textbf{A}}}^{-1}=
\mbox{\textit{\textbf{A}}}^{\mbox{\footnotesize {T}}}$.

The name comes from the fact that the matrix part of any isometry is an orthogonal matrix if referred to an orthonormal basis. In crystallography most matrices of the crystallographic symmetry operations are orthogonal if referred to the conventional basis.

Procedure: One forms the transposed matrix A $^{\mbox{\footnotesize {T}}}$ from the given matrix A and tests if it obeys the equation $\mbox{\textit{\textbf{A}}}\,\mbox{\textit{\textbf{A}}}^{\mbox{\footnotesize {T}}}
=\mbox{\textit{\textbf{I}}}$. If it does then the inverse $\mbox{\textit{\textbf{A}}}^{-1}=
\mbox{\textit{\textbf{A}}}^{\mbox{\footnotesize {T}}}$ is found. If not one has to go the general way.

There are several general methods to invert a matrix. Here we use a formula based on determinants. It is not restricted to dimensions 2 or 3.

Let A = $(A_{ik})$ be the matrix to be inverted, $\det(\mbox{\textit{\textbf{A}}})$ its determinant, and A$^{-1}$ = $((A^{-1})_{ik})$ be the inverted matrix which is to be determined. The coefficient $(A^{-1})_{ik}$ is determined from the equation

\begin{displaymath}(A^{-1})_{ik}=(\det(\mbox{\textit{\textbf{A}}}))^{-1}\,(-1)^{i+k}\,
\mbox{\textit{\textbf{B}}}_{ki},
\end{displaymath} (2.6.1)

where $\mbox{\textit{\textbf{B}}}_{ki}$ is that determinant which is obtained from $\det(\mbox{\textit{\textbf{A}}})$ by canceling the $k$-th row and $i$-th column. If $\det(\mbox{\textit{\textbf{A}}})$ is a $(2\times 2)$ determinant, then $B_{ki}$ is a number; if $\det(\mbox{\textit{\textbf{A}}})$ is a $(3\times 3)$ determinant, then $\mbox{\textit{\textbf{B}}}_{ki}$ is a $(2\times 2)$ determinant. In general, if $\det(\mbox{\textit{\textbf{A}}})$ is an $(n\times n)$ determinant, then B$_{ki}$ is an ( $(n-1)\times (n-1)$) determinant.

Note that in this equation the indices of $\mbox{\textit{\textbf{B}}}_{\mbox{\textit{\textbf{ki}}}}$ are exchanged with respect to the element $(A^{-1})_{\mbox{\textit{\textbf{ik}}}}$ which is to be determined.

Example. Calculate the inverse matrix of \( \mbox{\textit{\textbf{A}}} =
\left(\begin{array}{rrr} 1 & 2 & 0 \\
\bar{1} & 0 & 3 \\ 2 & \bar{1}& 0 \end{array} \right). \)

One determines $\det(\mbox{\textit{\textbf{A}}})=2\cdot3\cdot2-1\cdot3\cdot(-1)=15$ and obtains for the coefficients of A$^{-1}$

\( \begin{array}{ll}
(A^{-1})_{11} = (-1)^2 \left\vert \begin{array}{rr} 0 & 3 ...
...ay}{rr} 1 & 2 \\
\bar{1} & 0 \end{array} \right\vert/15 = 2/15.
\end{array} \)

With these coefficients one finds \(\mbox{\textit{\textbf{A}}}^{-1}=\left( \begin{array}{rrr}
1/5 & 0 & 2/5 \\ 2/5 & 0 & -1/5 \\ 1/15 & 1/3 & 2/15 \end{array} \right) \)

and verifies that $\mbox{\textit{\textbf{A}}}\,\mbox{\textit{\textbf{A}}}^{-1}=\mbox{\textit{\textbf{A}}}^{-1}\,\mbox{\textit{\textbf{A}}}=\mbox{\textit{\textbf{I}}}$ holds.


Distances and angles

In Section 1.6 formulae for the distance between points (by calculating the length of a vector) and the angle between bindings (vectors) have been derived. The scalar products of the basis vectors have been designated by $G_{ik}$, $i, k=1, 2, 3$. They form the

fundamental matrix of the coordinate basis \( \mbox{\textit{\textbf{G}}}= \left( \begin{array}{lll} G_{11} & G_{12} & G_{13} \\ G_{21} &
G_{22} & G_{23} \\ G_{31} & G_{32} & G_{33} \end{array} \right). \)

Because of $G_{ik}=G_{ki}$, G is a symmetric matrix.

In the formulae of Section 1.6 one may replace the `index formalism' by the `matrix formalism'. Using matrix multiplication with rows and columns,

\begin{displaymath}\mbox{ one obtains the formula for the distance \ }
r^2=\mbo...
...extbf{G}}}\,\mbox{\textit{\textbf{r}}}, \hspace{2em}\mbox{with}\end{displaymath}


\begin{displaymath}\mbox{\textit{\textbf{G}}} = \left( \begin{array}{ccc} a_1^2 ...
...\cos\beta & a_2\,a_3\,\cos\alpha &
a_3^2 \end{array} \right).
\end{displaymath} (2.6.2)

This is the same as equation (1.6.3) but expressed in another way. Such `matrix formulae' are useful in general calculations when changing the basis, when describing the relation between crystal lattice and reciprocal lattice, etc. However, for the actual calculation of distances, angles, etc. as well as for computer programs, the `index formulae' of Section 1.6 are more appropriate.

For orthonormal bases, because of G = I equation (2.6.2) becomes very simple:

\begin{displaymath}r^2=\mbox{\textit{\textbf{r}}}^{\mbox{\footnotesize {T}}}\,\mbox{\textit{\textbf{r}}}.
\end{displaymath} (2.6.3)

The formula for the angle $\Phi$ between the vectors ( $\stackrel{\longrightarrow}{SX}$) = r and ( $\stackrel{\longrightarrow}{SY}$) = t

\begin{displaymath}
% latex2html id marker 5965
\mbox{is \ }\ r\,t\,
\cos\Phi=\m...
...xtit{\textbf{t}}}\mbox{, \ see Fig. \ref{angl},
\hspace{1em}or}\end{displaymath}



\begin{displaymath}\cos\Phi=(\mbox{\textit{\textbf{r}}}^{\mbox{\footnotesize {T}...
...T}}}
\,\mbox{\textit{\textbf{G}}}\,\mbox{\textit{\textbf{t}}}.
\end{displaymath} (2.6.4)


The volume of the unit cell

The volume $V$ of the unit cell of a crystal structure, i.e. the body containing all points with coordinates $0\le x_1, x_2, x_3 <1$, can be calculated by the formula

\begin{displaymath}\det(\mbox{\textit{\textbf{G}}}) = V^2.
\end{displaymath} (2.6.5)

In the general case one obtains

\begin{displaymath}\rule{2em}{0ex} V^2 = \left\vert \begin{array}{ccc}
G_{11} &...
...& G_{23} \\ G_{31} & G_{32} & G_{33} \end{array} \right\vert = \end{displaymath}

\begin{displaymath}= a^2\,b^2\,c^2\,(1-\cos^2\alpha -\cos^2\beta
-\cos^2\gamma+2\cos\alpha\cos\beta\cos\gamma).
\end{displaymath} (2.6.6)

The formula (2.6.6) becomes simpler depending on the crystallographic symmetry, i.e. on the crystal system.


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