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A Generalized Solution of the Orthogonal Procrustes Problem

Published online by Cambridge University Press:  01 January 2025

Peter H. Schönemann
Peter H. Schönemann*
Affiliation:
Psychometric Laboratory, University of North Carolina†
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Abstract

A solutionT of the least-squares problem AT=B + E, given A and B so that trace (E′E)= minimum and T′T= I is presented. It is compared with a less general solution of the same problem which was given by Green [5]. The present solution, in contrast to Green's, is applicable to matrices A and B which are of less than full column rank. Some technical suggestions for the numerical computation of T and an illustrative example are given.

Information

Type
Original Paper
Information
Psychometrika , Volume 31 , Issue 1 , March 1966 , pp. 1 - 10
Copyright
Copyright © 1966 Psychometric Society

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Footnotes

*

This paper is based on parts of a thesis submitted to the Graduate College of the University of Illinois in partial fulfillment of the requirements for a Ph.D. degree in Psychology.

†

The work reported here was carried out while the author was employed by the Statistical Service Unit Research, U. of Illinois. It is a pleasure to express my appreciation to Prof. K. W. Dickman, director of this unit, for his continuous support and encouragement in this and other work. I also gratefully acknowledge my debt to Prof. L. Humphreys for suggesting the problem and to Prof. L. R. Tucker, who derived (1.7) and (1.8) in summation notation, suggested an iterative solution (not reported here) and who provided generous help and direction at all stages of the project.

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