Orthogonalization of impure rotations

Diamond, R.

doi:10.1107/97809553602060000561

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

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International Tables for Crystallography (2006). Vol. B. ch. 3.3, p. 367 | 1 | 2 |

Section 3.3.1.2.3. Orthogonalization of impure rotations

R. Diamond^a ^*

^aMRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England
Correspondence e-mail: rd10@cam.ac.uk

3.3.1.2.3. Orthogonalization of impure rotations

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There are several ways of deriving a strictly orthogonal matrix from a given approximately orthogonal matrix, among them the following.

(i) The Gram–Schmidt process. This is probably the simplest and the easiest to compute. If the given matrix consists of three column vectors $[{\bf v}_{1}, {\bf v}_{2}]$ and $[{\bf v}_{3}]$ (later referred to as primers) which are to be replaced by three column vectors $[{\bf u}_{1}, {\bf u}_{2}]$ and $[{\bf u}_{3}]$ then the process is $[\eqalign{{\bf u}_{1} &= {\bf v}_{1}/|{\bf v}_{1}|\cr {\bf u}_{2} &= {\bf v}_{2} - ({\bf u}_{1} \cdot {\bf v}_{2}) {\bf u}_{1}\cr {\bf u}_{2} &= {\bf u}_{2}/|{\bf u}_{2}|\cr {\bf u}_{3} &= {\bf v}_{3} - ({\bf u}_{1} \cdot {\bf v}_{3}) {\bf u}_{1} - ({\bf u}_{2} \cdot {\bf v}_{3}) {\bf u}_{2}\cr {\bf u}_{3} &= {\bf u}_{3}/|{\bf u}_{3}|.}]$

As successive vectors are established, each vector v has subtracted from it its components in the directions of established vectors, and the remainder is normalized. The method will fail at the normalization step if the vectors v are not linearly independent. Otherwise, the process may be extended to any number of dimensions.

The weakness of the method is that, though $[{\bf u}_{1}]$ differs from $[{\bf v}_{1}]$ only in scale, $[{\bf u}_{N}]$ may differ grossly from $[{\bf v}_{N}]$ as the various columns are not treated equivalently.
(ii) A preferable method which treats all vectors equivalently is to iteratively replace the matrix M by $[{\textstyle{1\over 2}} ({\bi M} + {\bi M}^{T-1})]$ .

Defining the residual matrix E as $[{\bi E} = {\bi MM}^{T} - {\bi I},]$ then on each iteration E is replaced by $[{\bi E}^{2} ({\bi MM}^{T})^{-1}/4]$ and convergence necessarily ensues.
(iii) A third method resolves M into its symmetric and antisymmetric parts $[{\bi S} = {\textstyle{1\over 2}} ({\bi M} + {\bi M}^{T}),\quad {\bi A} = {\textstyle{1\over 2}} ({\bi M} - {\bi M}^{T}),\quad {\bi M} = {\bi S} + {\bi A}]$ and constructs an orthogonal matrix for which only S is altered. A determines l, m, n and θ as shown in Section 3.3.1.2.1, and from these a new S may be constructed.
(iv) A fourth method is to treat the general matrix M as a combination of pure strain and pure rotation. Setting $[{\bi M} = {\bi RT}]$ with R orthogonal and T symmetrical gives $[{\bi T} = ({\bi M}^{T} {\bi M})^{1/2}, \quad {\bi R} = {\bi M} ({\bi M}^{T} {\bi M})^{-1/2}.]$

The rotation so found is the one which exactly superposes those three mutually perpendicular directions which remain mutually perpendicular under the transformation M.

$[{\bi T} - {\bi I}]$ is then the strain tensor of an unrotated body.

Writing $[{\bi M} = {\bi TR}]$ , $[{\bi T} = ({\bi MM}^{T})^{1/2}]$ , $[{\bi R} = ({\bi MM}^{T})^{-1/2} {\bi M}]$ may also be useful, in which $[{\bi T} - {\bi I}]$ is the strain tensor of a rotated body. See also Section 3.3.1.2.2 (iv).