Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-01-08T11:58:44.491Z Has data issue: false hasContentIssue false
  • English
  • Français

On Two-Sided Orthogonal Procrustes Problems

Published online by Cambridge University Press:  01 January 2025

Peter H. Schönemann
Peter H. Schönemann*
Affiliation:
Ohio State University
Get access Rights & Permissions [Opens in a new window]

Abstract

A least squares method for approximating a given symmetric matrix B by another matrix B which is orthogonally similar to a second given matrix A is derived and then generalized to nonsymmetric (but square) A and B. A possible application to ordering problems is discussed.

Type
Original Paper
Information
Psychometrika , Volume 33 , Issue 1 , March 1968 , pp. 19 - 33
Copyright
Copyright © 1968 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

This work was partially supported as project No. 083 by the Ohio State University, College of Education Research Grants and Leaves Program and subsequently as project I RO3 MH14097-01 by the National Institute of Mental Health. Free computer time was provided by the Ohio State University Computer Center. The writer is also pleased to record his appreciation for the time and assistance awarded to him most generously by Professors D. R. Whitney and J. S. Rustagi of the Ohio State University Mathematics Department.

References

Bellman, R. Introduction to matrix analysis, New York: McGraw-Hill, 1960.Google Scholar
Cattell, R. B. Factor analysis. An introduction and manual for the psychologists and social Scientists, New York: Harper & Bros., 1952.Google Scholar
Combs, C. H. A theory of data, New York: Wiley, 1964.Google Scholar
Dwyer, P. S. and MacPhail, M. S. Symbolic matrix derivatives. Annals of Mathematical Statistics, 1948, 19, 517–534.CrossRefGoogle Scholar
Eckart, C. and Young, G. The approximation of one matrix by another of lower rank. Psychometrika, 1936, 1, 211–218.CrossRefGoogle Scholar
Foa, U. G. The structure of interpersonal behavior in the dyad. In Criswell, , Solomon, , Suppes, (Eds.), Mathematical methods in small group processes, Stanford: Stanford University Press, 1962.Google Scholar
Green, B. F. The orthogonal approximation of an oblique structure in factor analysis. Psychometrika, 1952, 17, 429–440.Google Scholar
Greenberg, M. G. A method of successive cumulations for the scaling of pair comparison preference judgments. Psychometrika, 1965, 30, 441–448.CrossRefGoogle ScholarPubMed
Guttman, L. A new approach to factor analysis: The Radex. In Lazarsfeld, P. F. (Eds.), Mathematical thinking in the social sciences, Glencoe, Illinois: Free Press, 1954.Google Scholar
Hardy, G. H., Littlewood, T. E. and Polya, A. Inequalities, Cambridge: University Press, 1959.Google Scholar
Householder, A. S.and Young, A. Matrix approximation and latent roots. American Mathematical Monthly, 1938, 45, 165–171.CrossRefGoogle Scholar
Hurley, J. R. and Cattell, R. B. Producing direct rotation to test a hypothesized factor structure. Behavioral Science, 1962, 7, 258–262.CrossRefGoogle Scholar
Kaiser, H. F. Scaling a simplex. Psychometrika, 1962, 27, 155–162.CrossRefGoogle Scholar
Kruskal, J. B. Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 1964, 29, 1–27.CrossRefGoogle Scholar
Mukherjee, B. N. Derivation of likelihood-ratio tests for Guttman quasi-simplex covariance structures. Psychometria, 1966, 31, 97–123.CrossRefGoogle Scholar
Schönemann, P. H. A generalized solution of the orthogonal Procrustes problem. Psychometrika, 1966, 31, 1–10.CrossRefGoogle Scholar
Schönemann, P. H. On the formal matrix differentiation of traces and determinants, Chapel Hill: University of North Carolina Psychometric Laboratory, 1965.Google Scholar
Schönemann, P. H., Bock, R. D. and Tucker, L. R. Some notes on a theorem by Eckart and Young, Chapel Hill: University of North Carolina Psychometric Laboratory, 1965.Google Scholar
Shepard, R. N. The analysis of proximities: Multidimensional scaling with an unknown distance function. Psychometrika, 1962, 27, 125–139.CrossRefGoogle Scholar
Tryon, R. C. Identification of social areas by cluster analysis. University of California Publications in Psychology, 1955, 8, No. 5.Google Scholar
Tucker, L. R. A method for synthesis of factor analysis studies. A. G. O. Personnel Research. Section. Rep. No. 984, Department of the Army, 1951.CrossRefGoogle Scholar
Wrobleski, W. J. Extension of the Dwyer-MacPhail matrix derivative calculus with applications to estimation problems involving errors-in-variables and errors-in-equations. Technical Report, The University of Michigan, 1963.Google Scholar