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The Rigid Orthogonal Procrustes Rotation Problem

Published online by Cambridge University Press:  01 January 2025

Jos M. F. ten Berge
Jos M. F. ten Berge*
Affiliation:
University of Groningen
*
Requests for reprints should be sent to J.M.F. ten Berge, Heijmans Institute, Grote Kruisstraat 2/1, 9712 TS Groningen, The Netherlands. E-mail: j.m.f.ten.berge@rug.nl
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Abstract

The problem of rotating a matrix orthogonally to a best least squares fit with another matrix of the same order has a closed-form solution based on a singular value decomposition. The optimal rotation matrix is not necessarily rigid, but may also involve a reflection. In some applications, only rigid rotations are permitted. Gower (1976) has proposed a method for suppressing reflections in cases where that is necessary. This paper proves that Gower’s solution does indeed give the best least squares fit over rigid rotation when the unconstrained solution is not rigid. Also, special cases that have multiple solutions are discussed.

Keywords

orthogonal Procrustes rotationrigid rotationleast squares rotation
Type
Original Paper
Information
Psychometrika , Volume 71 , Issue 1 , March 2006 , pp. 201 - 205
Copyright
Copyright © 2006 The Psychometric Society

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Footnotes

The author is obliged to Henk Kiers and Alwin Stegeman for helpful comments on a previous draft.

References

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