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Approximating a Symmetric Matrix

Published online by Cambridge University Press:  01 January 2025

R. A. Bailey  and
J. C. Gower
R. A. Bailey
Affiliation:
Statistics Department, Rothamsted Experimental Station
J. C. Gower*
Affiliation:
Statistics Department, Rothamsted Experimental Station
*
Requests for reprints should be sent to J. C. Gower, Statistics Department, Rothamsted Experimental Station, Harpenden, Herts, AL5 2JQ, United Kingdom.
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Abstract

We examine the least squares approximation C to a symmetric matrix B, when all diagonal elements get weight w relative to all nondiagonal elements. WhenB has positivity p and C is constrained to be positive semi-definite, our main result states that, when w ≥1/2, then the rank of C is never greater than p, and when w ≤1/2 then the rank of C is at least p. For the problem of approximating a given n × n matrix with a zero diagonal by a squared-distance matrix, it is shown that the sstress criterion leads to a similar weighted least squares solution with w =(n+2)/4; the main result remains true. Other related problems and algorithmic consequences are briefly discussed.

Keywords

dimensionalityEckart-Youngleast-squaresmatrix approximationsstress
Type
Original Paper
Information
Psychometrika , Volume 55 , Issue 4 , December 1990 , pp. 665 - 675
Copyright
Copyright © 1990 The Psychometric Society

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