勷勤数学•专家报告
题 目：Similarity via transversal intersection of manifolds
报 告 人：李忠善(Zhongshan Li) 教授 (邀请人：尤利华)
Georgia State University
时 间：7月1日 15：00-16：00
地 点：数科院东楼401
报告人简介:
李忠善(Zhongshan Li)教授出生于中国兰州，现为美国Georgia State University（佐治亚州立大学）数学系终身正教授。研究方向包括组合矩阵理论、代数图论、矩阵理论应用等。
李忠善教授1983年毕业于兰州大学数学专业，获理学学士；1986年毕业于北京师范大学数学专业，获理学硕士学位；1990年毕业于North Carolina（北卡罗来纳）州立大学数学专业，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年起担任数学系研究生部主任，并于2010年至2024年任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。
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， 并应邀在北京大学、中科院系统所、清华大学、北京交大、北京师范大学、南开大学、复旦大学、同济大学、中国科技大、Emory大学、Wisconsin大学、Auburn大学、Tennessee大学、上海交大、华东师大、上海大学、华中师大、西安交大、兰州大学、山东大学、中国海洋大学、中北大学、电子科大、福州大学、哈尔滨工程大学、黑龙江大学、长沙理工大、湘潭大学等数十所高校作学术报告，在《American Mathematical Monthly》， 《Linear Algebra and Its Applications》，《SIAM J. on Discrete Mathematics》， 《J. Combin. Theory Ser. 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80余篇，并撰写了学术专著 《Handbook of Linear Algebra》中关于符号模式矩阵的一章。
摘 要：
Let $A$ be an $n\times n$ real matrix. As shown in the recent paper ``The bifurcation lemma for strong properties in the inverse eigenvalue problem of a graph'', Linear Algebra Appl. 648 (2022), 70--87, by S.M. Fallat, H.T. Hall, J.C.-H. Lin, and B.L. Shader, if the manifolds $ \{ G^{-1} A G : G\in \text{GL}(n, \mathbb R) \}$ and $Q(\text{sgn}(A))$ (consisting of all real matrices having the same sign pattern as $A$), both considered as embedded submanifolds of $\mathbb R^{n \times n}$, intersect transversally at $A$, then every superpattern of sgn$(A)$ also allows a matrix similar to $A$. Those authors say that the matrix $A$ has the nonsymmetric strong spectral property (nSSP) if $X = 0$ is the only matrix satisfying $A \circ X = 0$ and $AX^T - X^TA = 0, $ and show that the nSSP property of $A$ is equivalent to the above transversality. In this talk, this transversality property of $A$ is characterized using an alternative, more direct and convenient condition, called the similarity-transversality property (STP). Let $X=[x_{ij}]$ be a generic matrix of order $n$ whose entries are independent variables. The STP of $A$ is defined as the full row rank property of the Jacobian matrix of the entries of $AX-XA$ at the zero entry positions of $A$ with respect to the nondiagonal entries of $X$. This new approach makes it possible to take better advantage of the combinatorial structure of the matrix $A$, and provides theoretical foundation for constructing matrices similar to a given matrix while the entries have certain desired signs. Many results on matrices with the STP are presented. In particular, important classes of zero-nonzero patterns and sign patterns that require or allow this transversality property are identified. Examples illustrating many possible applications (such as diagonalizability, number of distinct eigenvalues, nilpotence, idempotence, semi-stability, the minimal polynomial, and rank) are provided. Several intriguing open problems are raised.
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