Skip to main content
SpringerLink
Log in

On two-sided orthogonal procrustes problems

  • Published:
Psychometrika Aims and scope Submit manuscript

Abstract

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

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

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.

    Google Scholar 

  • Eckart, C. and Young, G. The approximation of one matrix by another of lower rank.Psychometrika, 1936,1, 211–218.

    Google Scholar 

  • Foa, U. G. The structure of interpersonal behavior in the dyad. In Criswell, Solomon and 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.

    Google Scholar 

  • Guttman, L. A new approach to factor analysis: The Radex. in P. F. Lazarsfeld (Ed.),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.

    Google Scholar 

  • Hurley, J. R. and Cattell, R. B. Producing direct rotation to test a hypothesized factor structure.Behavioral Science, 1962,7, 258–262.

    Google Scholar 

  • Kaiser, H. F. Scaling a simplex.Psychometrika, 1962,27, 155–162.

    Google Scholar 

  • Kruskal, J. B. Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis.Psychometrika, 1964,29, 1–27.

    Google Scholar 

  • Mukherjee, B. N. Derivation of likelihood-ratio tests for Guttman quasi-simplex covariance structures.Psychometria, 1966,31, 97–123.

    Google Scholar 

  • Schönemann, P. H. A generalized solution of the orthogonal Procrustes problem.Psychometrika, 1966,31, 1–10.

    Google Scholar 

  • Schönemann, P. H. On the formal matrix differentiation of traces and determinants. Research Memorandum No. 27, 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. Research Memorandum No. 25, 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, 219–246.

    Google Scholar 

  • Tryon, R. C. Identification of social areas by cluster analysis.University of California Publications in Psychology, 1955,8, No. 5.

  • 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.

  • 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.

Download references

Author information

Authors and Affiliations

  1. Ohio State University, USA

    Peter H. Schönemann

Authors
  1. Peter H. Schönemann
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

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.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schönemann, P.H. On two-sided orthogonal procrustes problems. Psychometrika 33, 19–33 (1968). https://doi.org/10.1007/BF02289673

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02289673

Keywords

Search

Navigation