Essential sign change numbers of full sign pattern matrices
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...
| Main Authors: | , , , , , , , |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2016-06-01
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| Series: | Special Matrices |
| Subjects: | |
| Online Access: | http://www.degruyter.com/view/j/spma.2016.4.issue-1/spma-2016-0023/spma-2016-0023.xml?format=INT |
| _version_ | 1818716030536515584 |
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| author | Chen Xiaofeng Fang Wei Gao Wei Gao Yubin Jing Guangming Li Zhongshan Shao Yanling Zhang Lihua |
| author_facet | Chen Xiaofeng Fang Wei Gao Wei Gao Yubin Jing Guangming Li Zhongshan Shao Yanling Zhang Lihua |
| author_sort | Chen Xiaofeng |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-17T19:12:47Z |
| format | Article |
| id | doaj.art-1cece89564a7494389759f66989e9af2 |
| institution | Directory Open Access Journal |
| issn | 2300-7451 |
| language | English |
| last_indexed | 2024-12-17T19:12:47Z |
| publishDate | 2016-06-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Special Matrices |
| spelling | doaj.art-1cece89564a7494389759f66989e9af22022-12-21T21:35:49ZengDe GruyterSpecial Matrices2300-74512016-06-014110.1515/spma-2016-0023spma-2016-0023Essential sign change numbers of full sign pattern matricesChen Xiaofeng0Fang Wei1Gao Wei2Gao Yubin3Jing Guangming4Li Zhongshan5Shao Yanling6Zhang Lihua7College of Math and Stat, Chongqing Jiaotong University, Chongqing, ChinaSchool of Instrument & Electronics, North University of China, Shanxi, ChinaDept of Math and Stat, Georgia State University, Atlanta, GA 30302, USADept of Math, North University of China, Taiyuan, Shanxi, ChinaDept of Math and Stat, Georgia State University, Atlanta, GA 30302, USADept of Math and Stat, Georgia State University, Atlanta, GA 30302, USADept of Math, North University of China, Taiyuan, Shanxi, ChinaDept of Math and Stat, Georgia State University, Atlanta, GA 30302, USAA 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.http://www.degruyter.com/view/j/spma.2016.4.issue-1/spma-2016-0023/spma-2016-0023.xml?format=INTsign 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 |
| spellingShingle | Chen Xiaofeng Fang Wei Gao Wei Gao Yubin Jing Guangming Li Zhongshan Shao Yanling Zhang Lihua Essential sign change numbers of full sign pattern matrices Special Matrices 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 |
| title | Essential sign change numbers of full sign
pattern matrices |
| title_full | Essential sign change numbers of full sign
pattern matrices |
| title_fullStr | Essential sign change numbers of full sign
pattern matrices |
| title_full_unstemmed | Essential sign change numbers of full sign
pattern matrices |
| title_short | Essential sign change numbers of full sign
pattern matrices |
| title_sort | essential sign change numbers of full sign pattern matrices |
| topic | 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 |
| url | http://www.degruyter.com/view/j/spma.2016.4.issue-1/spma-2016-0023/spma-2016-0023.xml?format=INT |
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