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On the Least-Squares Orthogonalization of an Oblique Transformation

Published online by Cambridge University Press:  01 January 2025

W. A. Gibson
W. A. Gibson*
Affiliation:
Department of Army†
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Abstract

After proving a special case of a theorem stated by Eckart and Young, namely, that an oblique transformation G is the product of two different orthogonal transformations and an intervening diagonal, this note shows that the best fitting orthogonal approximation to G is obtained simply by replacing the intervening diagonal by the identity matrix. This result is shown to be identical with two earlier orthogonalizing procedures when G is of full rank. A multiplicity of solutions is shown for the case of a singular G.

Type
Original Paper
Information
Psychometrika , Volume 27 , Issue 2 , June 1962 , pp. 193 - 195
Copyright
Copyright © 1962 The Psychometric Society

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Footnotes

*

I am grateful to J. J. Mellinger for pointing out a flaw in a previous version of this paper.

†

Opinions expressed herein are those of the author, not necessarily those of the Army.

References

Eckart, C. and Young, G. The approximation of one matrix by another of lower rank. Psychometrika, 1936, 1, 211–218.CrossRefGoogle Scholar
Gibson, W. A. Gramian factoring. (Unpublished manuscript)Google Scholar
Gibson, W. A. Orthogonal and oblique simple structures. Psychometrika, 1952, 17, 317–323.CrossRefGoogle Scholar
Gibson, W. A. Orthogonal from oblique transformations. Educ. Psychol. Measmt, 1960, 20, 713–721.CrossRefGoogle Scholar
Green, B. F. The orthogonal approximation of an oblique structure in factor analysis. Psychometrika, 1952, 17, 429–440.CrossRefGoogle Scholar
Thurstone, L. L. Multiple-factor analysis, Chicago: Univ. Chicago Press, 1947.Google Scholar