Orthonormal frames of coordinates – rotation matrix

Authier, A.

doi:10.1107/97809553602060000628

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

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International Tables for Crystallography (2006). Vol. D. ch. 1.1, pp. 5-6

Section 1.1.2.3. Orthonormal frames of coordinates – rotation matrix

A. Authier^a ^*

^a Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France
Correspondence e-mail: aauthier@wanadoo.fr

1.1.2.3. Orthonormal frames of coordinates – rotation matrix

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An orthonormal coordinate frame is characterized by the fact that $[g_{ij} = \delta_{ij} \quad (= 0 \quad \hbox{if} \ \ i \neq j \ \hbox{ and } \ = 1 \ \hbox{ if } \ i = j). \eqno(1.1.2.6)]$ 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 $[\left. \matrix{A^{T} \ = A^{-1}\quad \Rightarrow\quad A^{T}A = I\cr B^{T} \ = B^{-1}\quad \Rightarrow\quad B^{T}B = I.\cr}\right\} \eqno(1.1.2.7)]$ The matrices A and B are unitary matrices or matrices of rotation and $[\Delta (A)^{2} = \Delta (B)^{2} = 1 \Rightarrow \Delta (A) = \pm 1. \eqno(1.1.2.8)]$

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.

References

International Tables for Crystallography (2006). Vol. D. ch. 1.1, pp. 5-6