Essential sign change numbers of full sign pattern matrices

Xiaofeng Chen; Wei Fang; Wei Gao; Yubin Gao; Guangming Jing; Zhongshan Li; Yanling Shao; Lihua Zhang

Special Matrices (2016)

  • Volume: 4, Issue: 1, page 247-254
  • ISSN: 2300-7451

Abstract

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A sign pattern (matrix) is a matrix whose entries are from the set {+, −, 0} and a sign vector is a vector whose entries are from the set {+, −, 0}. A sign pattern or sign vector is full if it does not contain any zero entries. 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. The notions of essential row sign change number and essential column sign change number are introduced for full sign patterns and condensed sign patterns. By inspecting the sign vectors realized by a list of real polynomials in one variable, a lower bound on the essential row and column sign change numbers is obtained. Using point-line confiurations on the plane, it is shown that even for full sign patterns with minimum rank 3, the essential row and column sign change numbers can differ greatly and can be much bigger than the minimum rank. Some open problems concerning square full sign patterns with large minimum ranks are discussed.

How to cite

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Xiaofeng Chen, et al. "Essential sign change numbers of full sign pattern matrices." Special Matrices 4.1 (2016): 247-254. <http://eudml.org/doc/281173>.

@article{XiaofengChen2016,
abstract = {A sign pattern (matrix) is a matrix whose entries are from the set \{+, −, 0\} and a sign vector is a vector whose entries are from the set \{+, −, 0\}. A sign pattern or sign vector is full if it does not contain any zero entries. 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. The notions of essential row sign change number and essential column sign change number are introduced for full sign patterns and condensed sign patterns. By inspecting the sign vectors realized by a list of real polynomials in one variable, a lower bound on the essential row and column sign change numbers is obtained. Using point-line confiurations on the plane, it is shown that even for full sign patterns with minimum rank 3, the essential row and column sign change numbers can differ greatly and can be much bigger than the minimum rank. Some open problems concerning square full sign patterns with large minimum ranks are discussed.},
author = {Xiaofeng Chen, Wei Fang, Wei Gao, Yubin Gao, Guangming Jing, Zhongshan Li, Yanling Shao, Lihua Zhang},
journal = {Special Matrices},
keywords = {sign pattern (matrix); full sign pattern; minimum rank; sign change number; essential row sign change number; essential column sign change number; sign vectors of a list of polynomials; sign pattern matrix; essential row sign change number},
language = {eng},
number = {1},
pages = {247-254},
title = {Essential sign change numbers of full sign pattern matrices},
url = {http://eudml.org/doc/281173},
volume = {4},
year = {2016},
}

TY - JOUR
AU - Xiaofeng Chen
AU - Wei Fang
AU - Wei Gao
AU - Yubin Gao
AU - Guangming Jing
AU - Zhongshan Li
AU - Yanling Shao
AU - Lihua Zhang
TI - Essential sign change numbers of full sign pattern matrices
JO - Special Matrices
PY - 2016
VL - 4
IS - 1
SP - 247
EP - 254
AB - A sign pattern (matrix) is a matrix whose entries are from the set {+, −, 0} and a sign vector is a vector whose entries are from the set {+, −, 0}. A sign pattern or sign vector is full if it does not contain any zero entries. 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. The notions of essential row sign change number and essential column sign change number are introduced for full sign patterns and condensed sign patterns. By inspecting the sign vectors realized by a list of real polynomials in one variable, a lower bound on the essential row and column sign change numbers is obtained. Using point-line confiurations on the plane, it is shown that even for full sign patterns with minimum rank 3, the essential row and column sign change numbers can differ greatly and can be much bigger than the minimum rank. Some open problems concerning square full sign patterns with large minimum ranks are discussed.
LA - eng
KW - sign pattern (matrix); full sign pattern; minimum rank; sign change number; essential row sign change number; essential column sign change number; sign vectors of a list of polynomials; sign pattern matrix; essential row sign change number
UR - http://eudml.org/doc/281173
ER -

References

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