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Retrieving the Correlation Matrix from a Truncated PCA Solution: The Inverse Principal Component Problem

Published online by Cambridge University Press:  01 January 2025

Jos M. F. ten Berge  and
Henk A. L. Kiers
Jos M. F. ten Berge*
Affiliation:
University of Groningen
Henk A. L. Kiers
Affiliation:
University of Groningen
*
Requests for reprints should be sent to Jos M.E ten Berge, Heijmans Institute, University of Groningen, Grote Kruisstraat 2/1, 9712 TS Groningen, THE NETHERLANDS. Email: [email protected]
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Abstract

When r Principal Components are available for k variables, the correlation matrix is approximated in the least squares sense by the loading matrix times its transpose. The approximation is generally not perfect unless r =k. In the present paper it is shown that, when r is at or above the Ledermann bound, r principal components are enough to perfectly reconstruct the correlation matrix, albeit in a way more involved than taking the loading matrix times its transpose. In certain cases just below the Ledermann bound, recovery of the correlation matrix is still possible when the set of all eigenvalues of the correlation matrix is available as additional information.

Keywords

Ledermann boundprincipal component analysisinverse eigenvalue problems
Type
Original Paper
Information
Psychometrika , Volume 64 , Issue 3 , September 1999 , pp. 317 - 324
Copyright
Copyright © 1999 The Psychometric Society

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