Date of Award
2-2-2006
Degree Type
Thesis
Degree Name
Master of Science (MS)
Department
Mathematics and Statistics
First Advisor
Frank J. Hall
Second Advisor
Zhongshan Li
Third Advisor
Marina Arav
Abstract
A sign pattern matrix is a matrix whose entries are from the set {+,-,0}. The minimum rank of a sign pattern matrix A is the minimum of the rank 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 - 1,(where A is mxn), 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 -1 or 1 are explored. Sign patterns that almost require unique rank are also investigated.
DOI
https://doi.org/10.57709/1059661
Recommended Citation
Koyuncu, Selcuk, "Rational Realizations of the Minimum Rank of a Sign Pattern Matrix." Thesis, Georgia State University, 2006.
doi: https://doi.org/10.57709/1059661