Brown Bag Seminar - March 7, 2006
Speaker: Marina Arav
Affiliation: Department of Mathematics and Statistics, Georgia State University
Title: Rational Realizations of the Minimum Rank of a Sign Pattern Matrix
Time: 12:00 - 1:00 PM
Location: RUTCOR Building - Lounge, Rutgers University, Busch Campus, Piscataway, NJ
Abstract:
A sign pattern matrix is a matrix whose entries come from the
set {+, -, 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 is 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 is least n-1 (where A is m by 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 1 or -1 are
investigated.
This is joint work with Frank Hall, Selcuk Koyuncu, Zhongshan Li, and
Bhaskara Rao.
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