Convex polytopes and minimum ranks of nonnegative sign pattern matrices

Title: Convex polytopes and minimum ranks of nonnegative sign pattern matrices

Speaker: Georgia State University 李忠善教授

时间: 2018/07/02   15:00

地点: 知新楼B1032

邀请人: 王光辉教授

Abstract: A sign pattern matrix (resp., nonnegative sign pattern matrix) is a matrix whose entries are from the set $\{+, -, 0\}$ (resp., $ \{ +, 0 \}$). The minimum rank (resp., rational minimum rank) of a sign pattern matrix $\cal A$ is the minimum of the ranks of the matrices (resp., rational matrices) whose entries have signs equal to the corresponding entries of $\cal A$. Using a correspondence between sign patterns with minimum rank $r\geq 2$ and point-hyperplane configurations in $\mathbb R^{r-1}$ 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 $(d+1)\times (d+1)$ 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 $m\times n$ 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.

李忠善教授简介: 现为美国Georgia State University(佐治亚州立大学)数学系终身正教授。研究兴趣包括组合矩阵理论、代数图论、矩阵理论应用等。李忠善教授曾多次应邀出席数学国际学术会议并报告论文,在《American Mathematical Monthly》,《Linear Algebra and Its Applications》, 《SIAM J. on Discrete Math.》,《Linear and Multilinear Algebra》, 《J. Combin. Theory Ser. B》, 《IEEE Transactions on Neural Networks and Learning Systems》 等重要国际学术期刊上发表论文60余篇,近五年发表20余篇学术论文,并出版学术专著《Handbook of Linear Algebra》中的一章。李忠善教授目前还担任美国《Mathematical Reviews》特约评论员,《JP Journal of Algebra, Number Theory and Application》杂志编委,美国数学会会员,国际线性代数学会会员等职务。08-09年和15-16任加拿大国家科学和工程研究委员会项目评审专家。