International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 |
International Tables for Crystallography (2013). Vol. D, ch. 1.1, pp. 5-6
Section 1.1.2.3. Orthonormal frames of coordinates – rotation matrix^{a}Institut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France |
An orthonormal coordinate frame is characterized by the fact that One deduces from this that the scalar product is written simply as
Let us consider a change of basis between two orthonormal systems of coordinates: Multiplying the two sides of this relation by , it follows that which can also be written, if one notes that variance is not apparent in an orthonormal frame of coordinates and that the position of indices is therefore not important, as
The matrix coefficients, , are the direction cosines of with respect to the basis vectors. Similarly, we have so that where ^{T} indicates transpose. It follows that so that The matrices A and B are unitary matrices or matrices of rotation and
One can write for the coefficients giving six relations between the nine coefficients . There are thus three independent coefficients of the matrix A.