Binary factorizations of the matrix of all ones

Tréfois, Maguy;Van Dooren, Paul;Delvenne, Jean-Charles
(2015) Linear Algebra and Its Applications — Vol. 468, p. 63-79 (2015)

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In this paper, we consider the problem of factorizing the n×n matrix Jn of all ones into the n×n binary matrices. We show that under some conditions on the factors, these are isomorphic to a row permutation of a De Bruijn matrix. Moreover, we consider in particular the binary roots of Jn, i.e. the binary solutions to Am=Jn. On the one hand, we prove that any binary root with minimum rank is isomorphic to a row permutation of a De Bruijn matrix whose row permutation is represented by a block diagonal matrix. On the other hand, we partially solve Hoffman's open problem of characterizing the binary solutions to A2=Jn by providing a characterization of the binary solutions to A2=Jn with minimum rank. Finally, we provide a class of roots which are isomorphic to a De Bruijn matrix.
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Tréfois, M., Van Dooren, P., & Delvenne, J.-C. (2015). Binary factorizations of the matrix of all ones. Linear Algebra and Its Applications, 468, 63-79. https://doi.org/10.1016/j.laa.2014.01.011 (Original work published 2015)