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Special aspects of the matrix formalism

The contents of this chapter serve two purposes:

1. To forge a link between geometry and calculations;
2. To provide the tools for coordinate changes.

The first point is described in the first two sections. The questions to be discussed are:

(i)

How can the matrix-column pair be obtained when the geometric meaning of the symmetry operation is known ?

(ii)

Given a matrix-column pair, what is its geometric meaning ?

The second point is a practical one. The complexity and amount of calculations depend strongly on the coordinate system of reference for the geometric actions. Therefore, it is advantageous to be flexible and free to choose for each calculation the optimal coordinate system. This means to change the coordinate system if necessary and to know what happens with the coordinates and the matrix-column pairs by such a change. In the last section of this chapter, partitioned into 3 subsections, coordinate changes will be treated in 3 steps: Origin shift, change of basis, and change of both, i.e. general coordinate changes.

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