报告题目 (Title)：Similarity via transversal intersection of manifolds (流形横向交与相似性)
报告人 (Speaker)：Zhongshan Li 教授（佐治亚州立大学）
报告时间 (Time)：2024年6月17 (周一) 16:00
报告地点 (Place)：校本部GJ303
邀请人(Inviter)：王卿文、谭福平
主办部门：理学院数学系
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.
