TY - JOUR
T1 - Rational realization of the minimum ranks of nonnegative sign pattern matrices
AU - Fang, Wei
AU - Gao, Wei
AU - Gao, Yubin
AU - Gong, Fei
AU - Jing, Guangming
AU - Li, Zhongshan
AU - Shao, Yanling
AU - Zhang, Lihua
N1 - Publisher Copyright:
© 2016, Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic.
PY - 2016/9/1
Y1 - 2016/9/1
N2 - 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.
AB - 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.
KW - condensed sign pattern
KW - convex polytope
KW - integer matrix
KW - minimum rank
KW - nonnegative sign pattern
KW - point-hyperplane configuration
KW - rational minimum rank
KW - rational realization
KW - sign pattern (matrix)
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U2 - 10.1007/s10587-016-0299-1
DO - 10.1007/s10587-016-0299-1
M3 - Article
AN - SCOPUS:84991329283
SN - 0011-4642
VL - 66
SP - 895
EP - 911
JO - Czechoslovak Mathematical Journal
JF - Czechoslovak Mathematical Journal
IS - 3
ER -