Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
9498182 | Linear Algebra and its Applications | 2005 | 15 Pages |
Abstract
A sign pattern matrix is a matrix whose entries are from the set {+, â, 0}. 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. It is conjectured that the minimum rank of every sign pattern matrix can be realized by a rational matrix. The equivalence of this conjecture to several seemingly unrelated statements are established. For some special cases, such as when A is entrywise nonzero, or the minimum rank of A is at most 2, or the minimum rank of A is at least n â 1 (where A is m Ã n), the conjecture is shown to hold. Connections between this conjecture and the existence of positive rational solutions of certain systems of homogeneous quadratic polynomial equations with each coefficient equal to either â1 or 1 are investigated.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Marina Arav, Frank J. Hall, Selcuk Koyuncu, Zhongshan Li, Bhaskara Rao,