Rational realizations of the minimum rank of a sign pattern matrix

Arav, Marina; Hall, Frank J.; Koyuncu, Selcuk; Li, Zhongshan; Rao, B. V. Raghavendra

doi:10.1016/j.laa.2005.05.001

Cited by 20 publications

(20 citation statements)

References 10 publications

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“…Conversely, suppose that for each subspace L ⊆ R n with dimension r there exists a rational subspace K ⊆ R n with sign(K) = sign(L). Let A be an n × m sign pattern matrix with mr(A) = r and let F ∈ Q(A) with rank(F ) = r. There exist an n × r matrix U and an The next theorem can be found in [1] and [19].…”

Section: Lemmamentioning

confidence: 99%

Minimum ranks of sign patterns via sign vectors and duality

Arav

Hall

et al. 2015

ELA

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A sign pattern matrix is a matrix whose entries are from the set {+, −, 0}. The minimum rank of a sign pattern matrix A is the minimum of the ranks of the real matrices whose entries have signs equal to the corresponding entries of A. It is shown in this paper that for any m×n sign pattern A with minimum rank n − 2, rational realization of the minimum rank is possible. This is done using a new approach involving sign vectors and duality. It is shown that for each integer n ≥ 9, there exists a nonnegative integer m such that there exists an n × m sign pattern matrix with minimum rank n − 3 for which rational realization is not possible. A characterization of m × n sign patterns A with minimum rank n − 1 is given (which solves an open problem in Brualdi et al.[6]), along with a more general description of sign patterns with minimum rank r, in terms of sign vectors of certain subspaces. A number of results on the maximum and minimum numbers of sign vectors of k-dimensional subspaces of R n are obtained. In particular, it is shown that the maximum number of sign vectors of 2-dimensional subspaces of R n is 4n + 1. Several related open problems are stated along the way.

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Section: Lemmamentioning

confidence: 99%

Minimum ranks of sign patterns via sign vectors and duality

Arav

Hall

et al. 2015

ELA

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“…. , k}), we define V j i to be a neighbourhood of c j , such that: 1 ∈ I and c j ∈ I. For any j = 2, .…”

Section: Notation and First Remarksmentioning

confidence: 99%

“…Obviously the set of the matrices with a given sign pattern can be thought as a general interval matrices whose entries are from {(0, +∞), (−∞, 0), [0]}. There are several papers studying the minimal and maximal rank of the matrices with a given sign pattern, see for instance [1], [2], [10], [21]. In particular, in [1] and [2] the authors proved that the minimum rank of the real matrices with a given sign pattern is realizable by a rational matrix in case this minumum is at most 2 or at least min{p, q} − 2.…”

Section: Introductionmentioning

confidence: 99%

Generalization of real interval matrices to other fields

Rubei

2019

ELA

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An interval matrix is a matrix whose entries are intervals in R. We generalize this concept, which has been broadly studied, to other fields. Precisely we define a rational interval matrix to be a matrix whose entries are intervals in Q. We prove that a (real) interval p × q matrix with the endpoints of all its entries in Q contains a rank-one matrix if and only if contains a rational rank-one matrix and contains a matrix with rank smaller than min{p, q} if and only if it contains a rational matrix with rank smaller than min{p, q}; from these results and from the analogous criterions for (real) inerval matrices, we deduce immediately a criterion to see when a rational interval matrix contains a rank-one matrix and a criterion to see when it is full-rank, that is, all the matrices it contains are full-rank. Moreover, given a field K and a matrix α α α whose entries are subsets of K, we describe a criterion to find the maximal rank of a matrix contained in α α α.2010 Mathematical Subject Classification: 15A99, 15A03, 65G40

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“…For this graph the minimum rank over the rationals is strictly more than that over the reals (this rank being 6). Note that in [1] it was shown that for every real matrix B of rank 2 there is a rational matrix F of rank 2 such that B and F have the same sign pattern.…”

Section: The Counterexamplementioning

confidence: 99%

The minimum rank problem: A counterexample

Kopparty¹,

Rao

2008

Linear Algebra and its Applications

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We provide a counterexample to a recent conjecture that the minimum rank of every sign pattern matrix can be realized by a rational matrix. We use one of the equivalences of the conjecture and some results from projective geometry. As a consequence of the counterexample we show that there is a graph for which the minimum rank over the reals is strictly smaller than the minimum rank over the rationals. We also make some comments on the minimum rank of sign pattern matrices over different subfields of R.

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Rational realizations of the minimum rank of a sign pattern matrix

Cited by 20 publications

References 10 publications

Minimum ranks of sign patterns via sign vectors and duality

Minimum ranks of sign patterns via sign vectors and duality

Generalization of real interval matrices to other fields

The minimum rank problem: A counterexample

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