### 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 R^{r−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 language | English (US) |
---|---|

Pages (from-to) | 44-62 |

Number of pages | 19 |

Journal | Linear Algebra and Its Applications |

Volume | 558 |

DOIs | |

State | Published - Dec 1 2018 |

### Fingerprint

### 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

*Linear Algebra and Its Applications*,

*558*, 44-62. https://doi.org/10.1016/j.laa.2018.08.019

**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.

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 558, pp. 44-62. https://doi.org/10.1016/j.laa.2018.08.019

}

TY - JOUR

T1 - Minimum ranks of sign patterns and zero–nonzero patterns and point–hyperplane configurations

AU - Fang, Wei

AU - Gao, Wei

AU - Gao, Yubin

AU - Gong, Fei

AU - Jing, Guangming

AU - Li, Zhongshan

AU - Shao, Yanling

AU - Zhang, Lihua

PY - 2018/12/1

Y1 - 2018/12/1

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.

KW - Condensed sign pattern

KW - Maximum rank

KW - Minimum rank

KW - Point–hyperplane configuration

KW - Rational minimum rank

KW - Sign pattern matrix

KW - Zero–nonzero pattern

UR - http://www.scopus.com/inward/record.url?scp=85051621691&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85051621691&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2018.08.019

DO - 10.1016/j.laa.2018.08.019

M3 - Article

AN - SCOPUS:85051621691

VL - 558

SP - 44

EP - 62

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

ER -