The generalized inverse and inverse structure

When a structure in one space is projected or mapped or otherwise described in another space or 'language', then the transformation is usually irreversible. In the case of linear transformations a generalized inverse matrix exists even if the transformation matrix is rectangular or singular. This inverse represents the best that can be done by way of reverse transformation. Typical crystallographic applications of this inverse are developed. Some are in practical computing, where failures due to singularity of a matrix can be avoided. Others are in the theoretical treatment of redundant axes, such as are usual in the description of hexagonal crystals using four indices and four axes. The idea of the inverse of a set of crystallographic axes, normally the reciprocal axes, can be extended to the concept of the inverse of any set of coordinates.