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On the Matrix Equation X′X = A

Published online by Cambridge University Press:  20 January 2009

John E. Maxfield  and
Henryk Minc
John E. Maxfield
Affiliation:
University of Florida, Gainesville, Florida.
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If X is a matrix with non-negative entries then X′X is positive semi-definite with non-negative entries. Conversely, if A is positive semi-definite then there exist matrices Y, not necessarily with non-negative entries, such that Y′Y = A. In the present paper we investigate whether, given a positive semidefinite matrix A with non-negative entries, the equation X′X = A has a solution X with non-negative entries. An equivalent statement of the problem is: Can a positive semi-definite matrix with non-negative entries be expressed as a sum of rank 1 positive semi-definite matrices with non-negative entries? We answer the question in the affirmative for n≦4 and quote the following example due to M.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 13 , Issue 2 , December 1962 , pp. 125 - 129
Copyright
Copyright © Edinburgh Mathematical Society 1962

References

REFERENCE

(1) Hall, Marshall Jr., A Survey of Combinatorial Analysis, Surveys in Applied Mathematics IV (Wiley, 1958), 35–104.Google Scholar