Minimum Rank and Cycle Conditions for Sign Patterns That Allow Diagonalizability
报告题目: Minimum Rank and Cycle Conditions for Sign Patterns That Allow Diagonalizability
报告时间：2019年7月8日，星期一，下午16:00-18:00
报告地点：北五楼427会议室
报告人：李忠善教授，美国佐治亚州立大学
报告摘要:
A sign pattern (matrix) is a matrix whose entries are from the set $\{+, -, 0 \}$. A square sign pattern $\cal A$ is said to allow diagonalization if there is a diagonalizable real matrix whose entries have signs specified by the corresponding entries of $\cal A$. It is known that for every sign pattern that allows diagonalization, its maximum composite cycle length is greater than or equal to its minimum rank. It is also known that a sign pattern allows diagonalization if and only if it allows rank-principality. Characterization of sign patterns that allow diagonalization has been a long-standing open problem. In this talk, we establish some new necessary/sufficient conditions for a sign pattern to allow diagonalization, and explore possible ranks of diagonalizable matrices with a specified sign pattern. In particular, it is shown that every irreducible sign pattern with minimum rank 2 allows diagonalization at rank 2 and also at the maximum rank. Sign patterns whose maximal zero submatrices are ``strongly disjoint'' are shown to have a composite cycle consisting of 1-cycles, 2-cycles, and at most one 3-cycle, with total length equal to the maximum rank; for such sign patterns, the maximum composite cycle length is invariant under row and column permutations.
李忠善（Zhongshan Li）教授简介：
李忠善教授1983年毕业于兰州大学数学专业，获理学学士学位；1986年毕业于北京师范大学数学系，获理学硕士学位；1990年毕业于North Carolina（北卡罗来纳）州立大学，获理学博士学位。 自1991年起在美国乔治亚州立大学数学与统计系任教, 1998年成为Georgia（佐治亚）州立大学副教授及终身教授, 2007年晋升为正教授。2010年至2015年担任数学系研究生部主任，并于2010年成为佐治亚州立大学科学与艺术学院职称和终身教授评定委员会的成员（2017年起任此委员会的主席）。目前主要从事组合矩阵论的研究，包括符号模式矩阵、最小秩问题、非负矩阵、代数图论、整数矩阵、矩阵方程的有理解、实线性子空间的符号向量集等。曾多次应邀出席数学国际学术会议并作学术报告，在《American Mathematical Monthly》,《Linear Algebra and Its Applications》,《SIAM Journal on Discrete Mathematics》,《Journal of Combinational Theory Series B》,《Linear and Multilinear Algebra》,《Graphs and Combinatorics》,《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21篇学术论文，主持或参与多项科研项目。现担任美国《Mathematical Reviews》特约评论员，《JP Journal of Algebra,Number Theory and Applications》杂志编委等职务。
