International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

International Tables for Crystallography (2006). Vol. A, ch. 5.1, p. 78

Section 5.1.2. Matrix notation

H. Arnolda

aInstitut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany

5.1.2. Matrix notation

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Throughout this volume, matrices are written in the following notation:

As ([{\it 1 \times 3}]) row matrices:
(a, b, c) the basis vectors of direct space
(h, k, l) the Miller indices of a plane (or a set of planes) in direct space or the coordinates of a point in reciprocal space
As ([{\it 3 \times 1}]) or ([{\it 4 \times 1}]) column matrices:
[x = (x/y/z)] the coordinates of a point in direct space
[({\bf a}^{*}/{\bf b}^{*}/{\bf c}^{*})] the basis vectors of reciprocal space
([u/v/w]) the indices of a direction in direct space
[{\bi p} = (p_{1}/p_{2}/p_{3})] the components of a shift vector from origin O to the new origin O
[{\bi q} = (q_{1}/q_{2}/q_{3})] the components of an inverse origin shift from origin O′ to origin O, with [{\bi q} = - {\bi P}^{-1}{\bi p}]
[{\bi w} = (w_{1}/w_{2}/w_{3})] the translation part of a symmetry operation [\hbox{\sf W}] in direct space
[\specialfonts{\bbsf x} = (x/y/z/1)] the augmented [(4 \times 1)] column matrix of the coordinates of a point in direct space
As ([{\it 3 \times 3}]) or ([{\it 4 \times 4}]) square matrices:
[{\bi P}, {\bi Q} = {\bi P}^{-1}] linear parts of an affine transformation; if P is applied to a [(1 \times 3)] row matrix, Q must be applied to a [(3 \times 1)] column matrix, and vice versa
W the rotation part of a symmetry operation [\hbox{\sf W}] in direct space
[\specialfonts{\bbsf P} = \pmatrix{{\bi P} &{\bi p}\cr {\bi o} &1\cr}] the augmented affine [(4 \times 4)] trans-formation matrix, with [{\bi o} = (0,0,0)]
[\specialfonts{\bbsf Q} = \pmatrix{{\bi Q} &{\bi q}\cr {\bi o} &1\cr}] the augmented affine [(4 \times 4)] trans-formation matrix, with [\specialfonts{\bbsf Q} = {\bbsf P}^{-1}]
[\specialfonts{\bbsf W} = \pmatrix{{\bi W} &{\bi w}\cr {\bi o} &1\cr}] the augmented [(4 \times 4)] matrix of a symmetry operation in direct space (cf. Chapter 8.1[link] and Part 11[link] ).








































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