Rational realization of the minimum ranks of nonnegative sign pattern matrices

Title:

Rational realization of the minimum ranks of nonnegative sign pattern matrices

Creator:

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

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:e9e0a487-0260-48f3-980b-fce5de7b2ed5
uuid:e9e0a487-0260-48f3-980b-fce5de7b2ed5
issn:0011-4642

Subject:

matematika, mathematics, sign pattern (matrix), nonnegative sign pattern, minimum rank, convex polytope, rational minimum rank, rational realization, integer matrix, condensed sign pattern, point-hyperplane configuration, 13, and 51

Type:

model:article and TEXT

Format:

print, bez média, and svazek

Description:

A sign pattern matrix (or nonnegative sign pattern matrix) is a matrix whose entries are from the set {+,−, 0} ({+, 0}, respectively). The minimum rank (or rational minimum rank) of a sign pattern matrix A is the minimum of the ranks of the matrices (rational matrices, respectively) whose entries have signs equal to the corresponding entries of A. Using a correspondence between sign patterns with minimum rank r ≥ 2 and point-hyperplane configurations in Rr−1 and Steinitz’s theorem on the rational realizability of 3-polytopes, it is shown that for every nonnegative sign pattern of minimum rank at most 4, the minimum rank and the rational minimum rank are equal. But there are nonnegative sign patterns with minimum rank 5 whose rational minimum rank is greater than 5. It is established that every d-polytope determines a nonnegative sign pattern with minimum rank d + 1 that has a (d + 1) × (d + 1) triangular submatrix with all diagonal entries positive. It is also shown that there are at most min{3m, 3n} zero entries in any condensed nonnegative m × n sign pattern of minimum rank 3. Some bounds on the entries of some integer matrices achieving the minimum ranks of nonnegative sign patterns with minimum rank 3 or 4 are established., Wei Fang, Wei Gao, Yubin Gao, Fei Gong, Guangming Jing, Zhongshan Li, Yanling Shao, Lihua Zhang., and Obsahuje seznam literatury

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

895-911

Source:

Czechoslovak Mathematical Journal | 2016 Volume:66 | Number:3

Harvested from:

CDK

Metadata only:

false