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The rigid orthogonal Procrustes rotation problem

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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.

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Authors and Affiliations

  1. University of Groningen, Groningen

    Jos M. F. ten Berge

  2. Heijmans Institute, Grote Kruisstraat 2/1, 9712 TS, Groningen, The Netherlands

    Jos M. F. ten Berge

Authors
  1. Jos M. F. ten Berge
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Correspondence to Jos M. F. ten Berge.

Additional information

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

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Berge, J.M.F.t. The rigid orthogonal Procrustes rotation problem. Psychometrika 71, 201–205 (2006). https://doi.org/10.1007/s11336-004-1160-5

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  • DOI: https://doi.org/10.1007/s11336-004-1160-5

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