组合数学讨论班

报告题目: Similarity via transversal intersection of manifolds

报告人:  Zhongshan Li, Georgia state University

时间: 2026年6月29日，10:30-11:30

地点: 海纳苑2幢303

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, the sign patterns that allow the STP are characterized, and important classes of zero-nonzero patterns and sign patterns that require 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.