Matrix algebra

Rossmann, M. G.; Arnold, E.

doi:10.1107/97809553602060000556

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 2.3, pp. 252-253 | 1 | 2 |

Section 2.3.6.2. Matrix algebra

M. G. Rossmann^a ^* and E. Arnold^b

^aDepartment of Biological Sciences, Purdue University, West Lafayette, Indiana 47907, USA, and ^bCABM & Rutgers University, 679 Hoes Lane, Piscataway, New Jersey 08854-5638, USA
Correspondence e-mail: mgr@indiana.bio.purdue.edu

2.3.6.2. Matrix algebra

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The initial step in the rotation-function procedure involves the orthogonalization of both crystal systems. Thus, if fractional coordinates in the first crystal system are represented by x, these can be orthogonalized by a matrix [β] to give the coordinates X in units of length (Fig. 2.3.6.2); that is, $[{\bf X} = [\boldbeta ]{\bf x}.]$ If the point X is rotated to the point X′, then $[{\bf X}' = [\boldrho ]{\bf X}, \eqno(2.3.6.6)]$ where $[\boldrho]$ represents the rotation matrix relating the two vectors in the orthogonal system. Finally, X′ is converted back to fractional coordinates measured along the oblique cell dimension in the second crystal by $[{\bf x}' = [\boldalpha ]{\bf X}'.]$ Thus, by substitution, $[{\bf x}' = [\boldalpha ] [\boldrho ]{\bf X} = [\boldalpha ] [\boldrho ] [\boldbeta ]{\bf x}, \eqno(2.3.6.7)]$ and by comparison with (2.3.6.2) it follows that $[[{\bi C}] = [\boldalpha ] [\boldrho ] [\boldbeta ].]$ Fig. 2.3.6.2 shows the mode of orthogonalization used by Rossmann & Blow (1962). With their definition it can be shown that $[[\boldalpha ] = \pmatrix{1 / (a_{1} \sin \alpha_{3} \sin \omega) &0 &0\cr 1 / (a_{2} \tan \alpha_{1} \tan \omega) &1 / a_{2} &-1 / (a_{2} \tan \alpha_{1})\cr - 1 / (a_{2} \tan \alpha_{3} \sin \omega) & &\cr -1 / (a_{3} \sin \alpha_{1} \tan \omega) &0 &1 / (a_{3} \sin \alpha_{1})\cr}]$ and $[[\boldbeta ] = \pmatrix{a_{1} \sin \alpha_{3} \sin \omega &0 &0\cr a_{1} \cos \alpha_{3} &a_{2} &a_{3} \cos \alpha_{1}\cr a_{1} \sin \alpha_{3} \cos \omega &0 &a_{3} \sin \alpha_{1}\cr},]$ where $[\cos \omega = (\cos \alpha_{2} - \cos \alpha_{1} \cos \alpha_{3}) / (\sin \alpha_{1} \sin \alpha_{3})]$ with $[0 \leq \omega \lt \pi]$ . For a Patterson compared with itself, $[[\boldalpha ] = [\boldbeta ]^{-1}]$ .

Figure 2.3.6.2 | top | pdf |

Relationships of the orthogonal axes $[X_{1}, X_{2}, X_{3}]$ to the crystallographic axes $[a_{1}, a_{2}, a_{3}]$ . [Reprinted from Rossmann & Blow (1962).]

Both spherical $[(\kappa, \psi, \varphi)]$ and Eulerian $[(\theta_{1}, \theta_{2}, \theta_{3})]$ angles are used in evaluating the rotation function. The usual definitions employed are given diagrammatically in Figs. 2.3.6.3 and 2.3.6.4. They give rise to the following rotation matrices.

Figure 2.3.6.3 | top | pdf |

Eulerian angles $[\theta_{1}, \theta_{2}, \theta_{3}]$ relating the rotated axes $[X'_{1}, X'_{2}, X'_{3}]$ to the original unrotated orthogonal axes $[X_{1}, X_{2}, X_{3}]$ . [Reprinted from Rossmann & Blow (1962).]

Figure 2.3.6.4 | top | pdf |

Variables ψ and φ are polar coordinates which specify a direction about which the axes may be rotated through an angle κ. [Reprinted from Rossmann & Blow (1962).]

(a) Matrix [ $[\boldrho]$ ] in terms of Eulerian angles $[\theta_{1}, \theta_{2}, \theta_{3}]$ : $[\pmatrix{- \sin \theta_{1} \cos \theta_{2} \sin \theta_{3} &\cos \theta_{1} \cos \theta_{2} \sin \theta_{3} &\sin \theta_{2} \sin \theta_{3}\cr + \cos \theta_{1} \cos \theta_{3} &+ \sin \theta_{1} \cos \theta_{3} &\cr \noalign{\vskip5pt} - \sin \theta_{1} \cos \theta_{2} \cos \theta_{3} &\cos \theta_{1} \cos \theta_{2} \cos \theta_{3} &\sin \theta_{2} \cos \theta_{3}\cr - \cos \theta_{1} \sin \theta_{3} &- \sin \theta_{1} \sin \theta_{3} &\cr \noalign{\vskip5pt} \ \sin \theta_{1} \sin \theta_{2}\hfill &\ - \cos \theta_{1} \sin \theta_{2}\hfill &\ \cos \theta_{2}\hfill\cr}]$ and (b) matrix [ $[\boldrho]$ ] in terms of rotation angle κ and the spherical polar coordinates ψ, φ: $[\displaylines{\left(\matrix{\cos \kappa + \sin^{2} \psi \cos^{2} \varphi (1 - \cos \kappa) &\sin \psi \cos \psi \cos \varphi(1 - \cos \kappa)\cr &+ \sin \psi \sin \varphi \sin \kappa\cr \noalign{\vskip5pt} \sin \psi \cos \psi \cos \varphi (1 - \cos \kappa) &\cos \kappa + \cos^{2} \psi (1 - \cos \kappa)\cr - \sin \psi \sin \varphi \sin \kappa &\cr \noalign{\vskip5pt} -\sin^{2} \psi \sin \varphi \cos \varphi (1 - \cos \kappa) &- \sin \psi \cos \psi \sin \varphi (1 - \cos \kappa)\cr -\cos \psi \sin \kappa &+ \sin \psi \cos \varphi \sin \kappa\cr}\right.\cr \noalign{\vskip5pt} \phantom{-\sin^{2} \psi \sin \varphi \cos \varphi (1 - \cos \kappa)} \left.\matrix{-\sin^{2} \psi \cos \varphi \sin \varphi(1 - \cos \kappa)\cr + \cos \psi \sin \kappa\cr \noalign{\vskip5pt} - \sin \psi \cos \psi \sin \varphi(1 - \cos \kappa)\cr -\sin \psi \cos \varphi \sin \kappa\cr \noalign{\vskip5pt} \cos \kappa + \sin^{2} \psi \sin^{2}\varphi (1 - \cos \kappa)\cr}\right)}]$ Alternatively, (b) can be expressed as $[\displaylines{\left(\matrix{\cos \kappa + u^{2} (1 - \cos \kappa) &uv (1 - \cos \kappa) - w \sin \kappa\cr vu (1 - \cos \kappa) + w \sin \kappa &\cos \kappa + v^{2} (1 - \cos \kappa)\cr wu (1 - \cos \kappa) - v \sin \kappa &wv (1 - \cos \kappa) + u \sin \kappa\cr}\right.\hfill\cr \noalign{\vskip5pt}\hfill \phantom{wu (1 - \cos \kappa) - v \sin \kappa} \left.\matrix{uw (1 - \cos \kappa) + v \sin \kappa\cr uw (1 - \cos \kappa) - u \sin \kappa\cr \cos \kappa + w^{2} (1 - \cos \kappa)\cr}\right),}]$ where u, v and w are the direction cosines of the rotation axis given by $[\eqalign{u &= \sin \psi \cos \varphi,\cr v &= \cos \psi,\cr w &= -\sin \psi \sin \varphi.}]$ This latter form also demonstrates that the trace of a rotation matrix is $[2 \cos \kappa + 1]$ .

The relationship between the two sets of variables established by comparison of the elements of the two matrices yields $[\eqalign{\cos (\kappa / 2) &= \cos (\theta_{2} / 2) \cos \left({\theta_{1} + \theta_{3} \over 2}\right),\cr \tan \varphi &= -\cot (\theta_{2} / 2) \sin \left({\theta_{1} + \theta_{3} \over 2}\right) \hbox{ sec } \left({\theta_{1} - \theta_{3} \over 2}\right),\cr \cos \varphi \tan \psi &= \cot \left({\theta_{1} - \theta_{3} \over 2}\right).}]$ Since φ and ψ can always be chosen in the range 0 to π, these equations suffice to find $[(\kappa, \psi, \varphi)]$ from any set $[(\theta_{1}, \theta_{2}, \theta_{3})]$ .

References

Rossmann, M. G. & Blow, D. M. (1962). The detection of sub-units within the crystallographic asymmetric unit. Acta Cryst. 15, 24–31.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 2.3, pp. 252-253