Bilbao Crystallographic Server

Matrix-Column Representations of Symmetry Operations

The matrix representation of a symmetry operation is given in its matrix-column pair: (W, w) where W is a 3x3 matrix called the rotational part or matrix part; and w is a 3x1 vector called the translation part or column part.

In crystallography in general, an efficient procedure is applied to condense the description of symmetry operations by matrix-column pairs considerably. The so-called short-hand notation of the matrix-column pair (W, w) consists of a coordinate triplet.

The following examples illustrate the assignments of the coordinate triplets to the matrix-column pairs:
  • The coordinate triplet of ITA y+1/2, -x+1/2, z+1/4 for the space group P43212 (No. 96). The linear and translational part of the symmetry operations are:

  • The matrix-column pair (W, w) of one of the symmetry operations of the space group P6522 (No. 179) is:

  • it can be represented in the short-hand notation by the coordinate triplet: -x+y, y, -z+1/2