Minimum ranks of sign patterns and zero–nonzero patterns and point–hyperplane configurations

Wei Fang, Wei Gao, Yubin Gao, Fei Gong, Guangming Jing, Zhongshan Li, Yanling Shao, Lihua Zhang

Research output: Contribution to journalArticle

Abstract

A sign pattern (matrix) is a matrix whose entries are from the set {+,−,0}. The minimum rank (respectively, rational minimum rank) of a sign pattern matrix A is the minimum of the ranks of the real (respectively, rational) matrices whose entries have signs equal to the corresponding entries of A. A sign pattern A is said to be condensed if A has no zero row or column and no two rows or columns are identical or negatives of each other. A zero–nonzero pattern (matrix) is a matrix whose entries are from the set {0,⋆}, where ⋆ indicates a nonzero entry. Many of the sign pattern notions carry over to zero–nonzero patterns, assuming that the ground field is R. In this paper, a direct connection between condensed m×n sign patterns and zero–nonzero patterns with minimum rank r and m point–n hyperplane configurations in Rr−1 is established. In particular, condensed sign patterns (and zero–nonzero patterns) with minimum rank 3 are closely related to point–line configurations on the plane. Using this connection, we construct the smallest known sign pattern whose minimum rank is 3 but whose rational minimum rank is greater than 3. It is proved that for any sign pattern or zero–nonzero pattern A, if the number of zero entries on each column of A is at most 2, then the rational and real minimum ranks of A are equal. Further, it is shown that for any zero–nonzero pattern A with minimum rank r≥3, if the number of zero entries on each column of A is at most r−1, then the rational minimum rank of A is also r. A few related conjectures and open problems are raised.

Original languageEnglish (US)
Pages (from-to)44-62
Number of pages19
JournalLinear Algebra and Its Applications
Volume558
DOIs
StatePublished - Dec 1 2018

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Minimum Rank
Sign Pattern
Configuration
Sign Pattern Matrix
Zero
Hyperplane
Open Problems

Keywords

  • Condensed sign pattern
  • Maximum rank
  • Minimum rank
  • Point–hyperplane configuration
  • Rational minimum rank
  • Sign pattern matrix
  • Zero–nonzero pattern

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

Cite this

Minimum ranks of sign patterns and zero–nonzero patterns and point–hyperplane configurations. / Fang, Wei; Gao, Wei; Gao, Yubin; Gong, Fei; Jing, Guangming; Li, Zhongshan; Shao, Yanling; Zhang, Lihua.

In: Linear Algebra and Its Applications, Vol. 558, 01.12.2018, p. 44-62.

Research output: Contribution to journalArticle

Fang, Wei ; Gao, Wei ; Gao, Yubin ; Gong, Fei ; Jing, Guangming ; Li, Zhongshan ; Shao, Yanling ; Zhang, Lihua. / Minimum ranks of sign patterns and zero–nonzero patterns and point–hyperplane configurations. In: Linear Algebra and Its Applications. 2018 ; Vol. 558. pp. 44-62.
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N2 - A sign pattern (matrix) is a matrix whose entries are from the set {+,−,0}. The minimum rank (respectively, rational minimum rank) of a sign pattern matrix A is the minimum of the ranks of the real (respectively, rational) matrices whose entries have signs equal to the corresponding entries of A. A sign pattern A is said to be condensed if A has no zero row or column and no two rows or columns are identical or negatives of each other. A zero–nonzero pattern (matrix) is a matrix whose entries are from the set {0,⋆}, where ⋆ indicates a nonzero entry. Many of the sign pattern notions carry over to zero–nonzero patterns, assuming that the ground field is R. In this paper, a direct connection between condensed m×n sign patterns and zero–nonzero patterns with minimum rank r and m point–n hyperplane configurations in Rr−1 is established. In particular, condensed sign patterns (and zero–nonzero patterns) with minimum rank 3 are closely related to point–line configurations on the plane. Using this connection, we construct the smallest known sign pattern whose minimum rank is 3 but whose rational minimum rank is greater than 3. It is proved that for any sign pattern or zero–nonzero pattern A, if the number of zero entries on each column of A is at most 2, then the rational and real minimum ranks of A are equal. Further, it is shown that for any zero–nonzero pattern A with minimum rank r≥3, if the number of zero entries on each column of A is at most r−1, then the rational minimum rank of A is also r. A few related conjectures and open problems are raised.

AB - A sign pattern (matrix) is a matrix whose entries are from the set {+,−,0}. The minimum rank (respectively, rational minimum rank) of a sign pattern matrix A is the minimum of the ranks of the real (respectively, rational) matrices whose entries have signs equal to the corresponding entries of A. A sign pattern A is said to be condensed if A has no zero row or column and no two rows or columns are identical or negatives of each other. A zero–nonzero pattern (matrix) is a matrix whose entries are from the set {0,⋆}, where ⋆ indicates a nonzero entry. Many of the sign pattern notions carry over to zero–nonzero patterns, assuming that the ground field is R. In this paper, a direct connection between condensed m×n sign patterns and zero–nonzero patterns with minimum rank r and m point–n hyperplane configurations in Rr−1 is established. In particular, condensed sign patterns (and zero–nonzero patterns) with minimum rank 3 are closely related to point–line configurations on the plane. Using this connection, we construct the smallest known sign pattern whose minimum rank is 3 but whose rational minimum rank is greater than 3. It is proved that for any sign pattern or zero–nonzero pattern A, if the number of zero entries on each column of A is at most 2, then the rational and real minimum ranks of A are equal. Further, it is shown that for any zero–nonzero pattern A with minimum rank r≥3, if the number of zero entries on each column of A is at most r−1, then the rational minimum rank of A is also r. A few related conjectures and open problems are raised.

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