Sign patterns with minimum rank 2 and upper bounds on minimum ranks

Li, Zhongshan; Gao, Yubin; Arav, Marina; Gao, Wei; Hall, Frank J.; Holst, Hein van der

doi:10.1080/03081087.2012.716431

Cited by 18 publications

(22 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Lemmamentioning

confidence: 99%

Minimum ranks of sign patterns via sign vectors and duality

Arav

Hall

et al. 2015

ELA

Self Cite

View full text Add to dashboard Cite

show abstract

“…The present paper establishes a connection of this problem with that of computing certain rank functions arisen from tropical geometry. We prove the conjecture on sign patterns of rational matrices formulated in [3], and we present the examples showing the optimality of our result.…”

Section: Introductionmentioning

confidence: 69%

“…The problem of constructing a matrix over a given ordered field with specified sign pattern and rank deserved a significant amount of attention in recent publications, see [3] and references therein. The present paper establishes a connection of this problem with that of computing certain rank functions arisen from tropical geometry.…”

Section: Introductionmentioning

confidence: 99%

Sign patterns of rational matrices with large rank

Shitov

2014

European Journal of Combinatorics

View full text Add to dashboard Cite

Let A be a real matrix. The term rank of A is the smallest number t of lines (that is, rows or columns) needed to cover all the nonzero entries of A. We prove a conjecture of Li et al. stating that, if the rank of A exceeds t − 3, there is a rational matrix with the same sign pattern and rank as those of A. We point out a connection of the problem discussed with the Kapranov rank function of tropical matrices, and we show that the statement fails to hold in general if the rank of A does not exceed t − 3.2000 Mathematics Subject Classification. 15A03, 15B35.

show abstract

“…The minimum ranks of sign pattern matrices have been the focus of a large number of papers (see e.g. [1], [2], [3], [4], [5], [6], [7], [9], [11], [12], [13], [18], [19], [20], [21], [22], [27]), and they have important applications in areas such as communication complexity [1], [24], [25], machine learning [14], neural networks [10], combinatorics [11], [16], [28], and discrete geometry [23].…”

Section: Introductionmentioning

confidence: 99%