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Properties and Applications of Gramian Factoring

Published online by Cambridge University Press:  01 January 2025

W. A. Gibson
W. A. Gibson*
Affiliation:
Queens College of the City University of New York
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Abstract

The Gramian factorization G of a Gramian R is square and symmetric and has no negative characteristic roots. It is shown to be that square factorization that is, in the least-squares sense, most isomorphic to R, most like a scalar K, and most highly traced, and to be the necessary and sufficient relation between the oblique vectors of an oblique transformation and the orthogonal vectors of the least-squares orthogonal counterpart. A slightly modified Gramian factorization is shown to be the factorization that is most isomorphic to a specified diagonal D, and to be the main part of an iterative procedure for obtaining “simplimax,” a square factor matrix with simple structure maximized in the sense of having the largest sum of squared diagonal loadings. Several published applications of Gramian factoring are cited.

Type
Original Paper
Information
Psychometrika , Volume 32 , Issue 4 , December 1967 , pp. 425 - 434
Copyright
Copyright © 1967 The Psychometric Society

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