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Another Remark on a Result of K. Goldberg

Published online by Cambridge University Press:  20 November 2018

Marvin Marcus
Marvin Marcus*
Affiliation:
University of California, Santa Barbara
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In [3] K. Goldberg showed that if A is a 0-1 matrix that satisfies

(1.1)

then for some permutation matrix P, PAP* is a direct sum of matrices each of which is either zero or consists only of ones. More recently J. L. Brenner [1] proved that if A ∦ 0 (i.e. A has non-negative entries) and satisfies (1) then there exists a permutation matrix P such that PAP* = A1⊕ … ⊕An in in which each Ai is either 0 or all positive, Ai > 0, and satisfies (1) as well.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 6 , Issue 1 , January 1963 , pp. 7 - 9
Copyright
Copyright © Canadian Mathematical Society 1963

References

1. Brenner, J.L., The matrix equation AA* = sA. Amer. Math. Monthly, v. 68, 9, (1961), p. 895.10.2307/2311691Google Scholar
2. Gantmacher, F.R., The Theory of Matrices, v. II. Chelsea Publishing Company, New York (1959).Google Scholar
3. Goldberg, K., The incidence equation AAT = sA. Amer, Math. Monthly, v. 67, (1960), p. 367.10.2307/2308983Google Scholar