  The crystallography of displacive phase transformations can be described with
three types of matrices: the lattice distortion matrix, the orientation
relationship matrix, and the correspondence matrix. The paper gives some
formula to express them in crystallographic bases, orthonormal bases, and
reciprocal bases, and it explains how to use them to deduce the matrices of
inverse transformation. In the case of hard-sphere assumption, a continuous
form of the distortion matrix can be determined, and its derivative is
identified to the velocity gradient used in continuum mechanics. The
distortion, the orientation and the correspondence variants are determined by
coset decomposition with intersection groups that depend on the point groups of
the phases and on the type of transformation matrix. The stretch variants
required in the phenomenological theory of martensitic transformation should be
distinguished from the correspondence variants. The orientation variants and
the correspondence variants are also different; they are defined from the
geometric symmetries and algebraic symmetries, respectively. The concept of
orientation (ir)reversibility during thermal cycling is briefly and partially
treated by generalizing the orientation variants with n-cosets and graphs. Some
simple examples are given to show that there is no general relation between the
numbers of distortion, orientation and correspondence variants, and to
illustrate the concept of orientation variants formed by thermal cycling.
