报告学者：Jason (Zhongshan) Li

报告者单位：Georgia state University

报告时间：2024年7月10日（周三）下午16:00-17:00

报告地点：思西507

报告摘要：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. 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.    

报告学者简介：Jason (Zhongshan) Li，美国Georgia State University（佐治亚州立大学）数学与统计系终身教授，博士生导师，数学系研究生部主任。1983年毕业于兰州大学，获数学学士学位；1986年毕业于北京师范大学，获数学硕士学位；1990年毕业于North Carolina State University（北卡罗莱纳州立大学），获得数学博士学位。研究领域包括组合矩阵论，多项式与矩阵方程、矩阵不等式、代数图论、神经网络等。在IEEE Transactions on Neural Networks and Learning Systems、Linear Algebra and Its Applications、SIAM Journal on Discrete Mathematics、Linear & Multilinear Algebra等国际重要学术期刊上发表论文80余篇，主持并参与多项科研项目。此外，还担任JP Journal of Algebra Number Theory and Applications 等期刊的编委。