报告题目：Convex polytopes and minimum ranks of nonnegative sign pattern matrices

报 告 人：李忠善教授（美国乔治亚州立大学）

报告时间：2019年6月19日  15:00-16:00

报告地点：数统院307学术报告厅

报告摘要：

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.

报告人简介：

李忠善，美国佐治亚州立大学数学系终身正教授。兰州大学数学学学士、北京师范大学硕士美国北卡罗来纳州立大学博士。研究兴趣包括组合矩阵理论、代数图论、矩阵理论应用等。