报告题目 (Title)：Latest developments on the rank-preserving transversality property (RPTP) (保秩横截性质 (RPTP) 的最新进展)
报告人 (Speaker)：Li Zhongshan 教授（佐治亚州立大学）
报告时间 (Time)：2026年6月19日 (周五) 15:00
报告地点 (Place)：校本部GJ303
邀请人(Inviter)：谭福平
主办部门：理学院数学系
报告摘要(Abstract)： Let $A$ be a $m \times n$ real matrix. If the manifolds ${\widetilde{\cal M}_A}= \{ G A H^{-1} : G, H \text{ are nonsingular } \}$ and $Q(\text{sgn}(A))$ intersect transversally at $A,$ that is, the tangent spaces of ${\widetilde{\cal M}_A}$ and $Q(\text{sgn}(A))$ at $A$ sum to $ \mathbb R ^{m\times n},$ we say that $A$ has the rank-preserving transversality property (RPTP) and that $A$ is an RPTP matrix. This talk presents many important properties of RPTP matrices. For example, it is shown that the RPTP matrices are closed under permutation equivalence, diagonal equivalence, and transpose. Further, it is shown that a block upper triangular matrix with all diagonal blocks square has the RPTP if and only if each diagonal block has the RPTP and at most one diagonal block is singular. Two stronger but simpler properties, the row-equivalence transversality property (RETP) and the column-equivalence transversality property (CETP), are introduced and used to study RPTP matrices. Several techniques for partitioning a matrix to more easily check if it has the RPTP are presented. Some classes of sign patterns and zero-nonzero patterns that require the RPTP are identified. Just as the Strong Spectral Property is useful in studying the spectra of symmetric matrices associated with a graph, the notion of RPTP is a useful tool for studying the minimum ranks of sign patterns and zero-nonzero patterns.
