Tables for
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 1.1, pp. 5-6

Section Orthonormal frames of coordinates – rotation matrix

A. Authiera*

aInstitut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France
Correspondence e-mail: Orthonormal frames of coordinates – rotation matrix

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An orthonormal coordinate frame is characterized by the fact that [g_{ij} = \delta_{ij} \quad (= 0 \quad \hbox{if} \ \ i \neq j \ \hbox{ and } \ = 1 \ \hbox{ if } \ i = j). \eqno(]One deduces from this that the scalar product is written simply as [{\bf x} \cdot {\bf y} = x^{i}y\hskip 2pt^{j}g_{ij} = x^{i}y^{i}.]

Let us consider a change of basis between two orthonormal systems of coordinates: [{\bf e}_{i} = A\hskip1pt_{i}^{j}{\bf e}'_{j}.]Multiplying the two sides of this relation by [{\bf e}'_{j}], it follows that [{\bf e}_{i} \cdot {\bf e}'_{j} = A\hskip1pt_{i}^{j} {\bf e}'_{k} \cdot e'_{j} = A\hskip1pt_{i}^{j} g'_{kj} = A\hskip1pt_{i}^{j} \delta_{kj} \ \hbox{ (written correctly),}]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 [{\bf e}_{i} \cdot {\bf e}'_{j} = A\hskip1pt_{i}^{j} \ \hbox{ (written incorrectly)}.]

The matrix coefficients, [A\hskip1pt_{i}^{j}], are the direction cosines of [{\bf e}'_{j}] with respect to the [{\bf e}_{i}] basis vectors. Similarly, we have [B_{j}^{i} = {\bf e}_{i} \cdot {\bf e}'_{j}]so that [A\hskip1pt_{i}^{j} = B_{j}^{i} \quad \hbox{or} \quad A = B^{T},]where T indicates transpose. It follows that [A = B^{T} \hbox{ and } A = B^{-1}]so that [\left. \matrix{A^{T} \ = A^{-1}\quad \Rightarrow\quad A^{T}A = I\cr B^{T} \ = B^{-1}\quad \Rightarrow\quad B^{T}B = I.\cr}\right\} \eqno(]The matrices A and B are unitary matrices or matrices of rotation and [\Delta (A)^{2} = \Delta (B)^{2} = 1 \Rightarrow \Delta (A) = \pm 1. \eqno(]

  • If [\Delta (A) = 1] the senses of the axes are not changed – proper rotation.

  • If [\Delta (A) = - 1] the senses of the axes are changed – improper rotation. (The right hand is transformed into a left hand.)

One can write for the coefficients [A\hskip1pt_{i}^{j}][A\hskip1pt_{i}^{j}B_{j}^{k} = \delta_{i}^{k};\quad A\hskip1pt_{i}^{j}A_{j}^{k} = \delta_{i}^{k},]giving six relations between the nine coefficients [A\hskip1pt_{i}^{j}]. There are thus three independent coefficients of the [3 \times 3] matrix A.

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