Peter H. Schönemann
Professor Emeritus • Department of Psychological Sciences • Purdue University

Schonemann, P. H.

A solution of the orthogonal procrustes problem with applications to orthogonal and oblique rotation.  Dissertation Abstracts, 1965, 25, 4810.


Introduction and Overview

Chapter I. The "orthogonal Procrustes" problem

1.1. Definition of the problem and solution
1.2 Sufficiency and uniqueness of the solution
1.3 Comparison with Green's results
1.4 Program and illustrative examples

Chapter II

2.1 Varisim
2.2 Comparison with Varimax
2.3 Program and illustrative examples

Chapter III

3.1 Oblisim
3.2 Illustrative examples
3.3 Applications to "Hierarchical" factor analysis
3.4 Schmid-Leiman back solutions
3.5 Program and illustrative example


Appendix A.1 Some notes on symbolic matrix derivatives of traces

Appendix A.2 Program listings

Appendix A.3 Illustrative examples



1. This is my Ph.D. thesis. The degree was conferred June 1964 by the University of Illinois. The work on the Orthogonal Procustes Problem was completed sometime late in 1963.

The basic problem is to find a least squares solution for T in B = AT + E,  for given A, B of the same order, but not necessarily of full column rank, so that  T is orthogonal (i.e., T'T = I).  Ledyard Tucker and Lloyd Humphreys suggested this problem to me. They were interested in the deficient rank case as it arises with Schmid-Leiman solutions. Tucker had worked out an iterative algorithm for  this problem.  Using matrix derivatives, I arrived at the algebraic solution, T = VW', if VDW' is the singular value decomposition of A'B.

2. Bert Green had previously treated to  the full rank case  (B. F. Green, The orthogonal approximation of an oblique structure in factor analysis, Psychometrika, 1952, 17, 429-440). Earlier work includes Ahmavaara, Y, Transformation analysis of factorial data. Helsinki: Annals Academiae Scientiarum Fennicae, 1957, Series B, 106, and Ahmavaara, Y., On the mathematical theory of transformation analysis. Report No. 1. Helsinki: Suomalaisen Kirjallisuden Kirjapairo Oy Helsinki, 1963. Ahmavaara was a theoretical physicist who  was led to such transformations by the principle of invariance,  a guiding meta-principle of the natural sciences.  These days, his work is rarely cited. For more recent developments, see: Borg I. and Groenen, P.: Modern Multidimensional Scaling. New York: Springer, 1997, p. 339f.

3. After receiving he degree I published Chapter I, with only minor revisions, as  A generalized solution of the orthogonal Procrustes Problem  in Psychometrika in 1966.

The same Psychometrika issue also contains an article by Norman Cliff (Orthogonal rotation to congruence, Psychometrika, 1966, 31, 33-42) which gives the same solution. In a footnote on p. 36, it refers to my Psychometrika paper, but not to my thesis which had appeard a year earlier. While working on my thesis in 1963, I was not apprised of any concurrent work by others. The first time I saw Cliff's paper was when it appeard in print  alongside mine.

4. These results were then applied to (a) Schmidt-Leiman factor patterns (which, by definition, are of deficient column rank), and (b) orthogonal rotation ("Varisim", cf. [4] in  Publications.

5.  Appendix (A.1) of the thesis contains the main results on matrix derivatives I had obtained by this time. I completed an extended version of this work, also covering determinants, during my 1964/65 postdoc at UNC. It appeared much later under the title On the formal differentiation to traces and determinants On the formal differentiation of traces and determinants in Multivariate Behavioral Research, 1985, 20, 113-139.