The combination of 2 symmetry operations follows the procedure of Section 2.2. In analogy to equations (2.2.5) to (2.2.8) one obtains
These equations may be formulated with matrix-column pairs:
By comparing equations (4.2.4) and (4.2.5) one obtains
The multiplication of matrix-column pairs is associative, because
and on the other hand,
By comparison of both expressions one finds
A linear mapping is a mapping which leaves the origin fixed. Its column
part is thus the o column. According to equation (4.2.6)
any matrix-column pair can be decomposed into a linear mapping (W,o)
containing W only and a translation (I,w) with w only:
Question: What is the result if the translation (I,w) is performed first, and the linear mapping (W,o) after that, i.e. if the factors are exchanged ?
Before the reversion of a symmetry operation is dealt with, a general remark is appropriate. In general, the formulae of this section are not restricted to crystallographic symmetry operations but are valid also for affine mappings. However, there is one exception. In the inversion of a matrix W the determinant appears in the denominator of the coefficients of , see Subsection 2.6.1. Therefore, the condition has to be fulfilled. Such mappings are called regular or non-singular. Otherwise, if , the mapping is a projection and can not be reverted. For crystallographic symmetry operations, i.e. isometries W, always holds. Therefore, an isometry is always reversible, a general affine mapping may not be. Projections are excluded from this manuscript because they do not occur in crystallographic groups.
Now to the calculation of the reverse of a matrix-column pair. It is often necessary to know which matrix C and column c belong to that symmetry operation C which makes the original action W undone, i.e. which maps every image point onto the original point . The operation C is called the reverse operation of W. The combination of W with C restores the original state and the combined action CW maps . It is the identity operation I which maps any point onto itself. The operation I is described by the matrix-column pair (I,o), where I is the unit matrix and o is the column consisting of zeroes only. This means
It is always good to test the result of a calculation or derivation. One verifies the validity of the equations by applying equations (4.2.6) and (4.2.12). In addition in the following Problem 2A the results of this section may be practised.
Problem 2A. Symmetry described by matrix-column pairs.
For the solution, see p. .
In Vol. A of International Tables for Crystallography the crystallographic symmetry operations A, B, ... are referred to a conventional coordinate system and are represented by matrix-column pairs (A,a), (B,b), .... Among others one finds in the space-group tables of IT A indirectly, see Section 4.6:
Combining two symmetry operations or reversion of a symmetry operation corresponds to multiplication or reversion of these matrix-column pairs, such that the resulting matrix-column pair represents the resulting symmetry operation.
The following calculations make use of the formulae 4.2.6 and
Can one exploit the fact that the matrices A, B, C, and D are orthogonal matrices ?
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