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The Approximation of One Matrix by Another of Lower Rank

Published online by Cambridge University Press:  01 January 2025

Carl Eckart  and
Gale Young
Carl Eckart
Affiliation:
University of Chicago, Chicago, Illinois
Gale Young
Affiliation:
University of Chicago, Chicago, Illinois
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Abstract

The mathematical problem of approximating one matrix by another of lower rank is closely related to the fundamental postulate of factor-theory. When formulated as a least-squares problem, the normal equations cannot be immediately written down, since the elements of the approximate matrix are not independent of one another. The solution of the problem is simplified by first expressing the matrices in a canonic form. It is found that the problem always has a solution which is usually unique. Several conclusions can be drawn from the form of this solution.

A hypothetical interpretation of the canonic components of a score matrix is discussed.

Type
Original Paper
Information
Psychometrika , Volume 1 , Issue 3 , September 1936 , pp. 211 - 218
Copyright
Copyright © 1936 The Psychometric Society

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