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Rank-Reducibility of a Symmetric Matrix and Sampling Theory of Minimum Trace Factor Analysis

Published online by Cambridge University Press:  01 January 2025

Alexander Shapiro*
Affiliation:
Ben-Gurion University of the Negev
*
Requests for reprints should be sent to A. Shapiro, Department of Statistics & O.R., University of South Africa, P.O. Box 392, PRETORIA 0001, South Africa.

Abstract

One of the intriguing questions of factor analysis is the extent to which one can reduce the rank of a symmetric matrix by only changing its diagonal entries. We show in this paper that the set of matrices, which can be reduced to rank r, has positive (Lebesgue) measure if and only if r is greater or equal to the Ledermann bound. In other words the Ledermann bound is shown to be almost surely the greatest lower bound to a reduced rank of the sample covariance matrix. Afterwards an asymptotic sampling theory of so-called minimum trace factor analysis (MTFA) is proposed. The theory is based on continuous and differential properties of functions involved in the MTFA. Convex analysis techniques are utilized to obtain conditions for differentiability of these functions.

Type
Original Paper
Copyright
Copyright © 1982 The Psychometric Society

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References

Reference Note

Della Riccia, G. & Shapiro, A. Minimum rank and minimum trace of covariance matrices, 1980, Beer-Sheva, Israel: Department of Mathematics, Ben-Gurion University of the Negev.Google Scholar

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