报告题目 (Title)：The orthogonal equivalence transversality property (正交等价横截相交性)
报告人 (Speaker)： Zhongshan Li 教授（佐治亚州立大学）
报告时间 (Time)：2025年10月17日 (周五) 9:00
报告地点 (Place)：腾讯会议号：175-906-172
邀请人(Inviter)：谭福平
主办部门：太阳集团官网8722数学系
报告摘要： Let $A$ be a $m \times n$ real matrix. If the manifolds ${\cal M}^{\approx}_A = \{ U A V^T \mid U \in {\cal O}_m, \; V \in {\cal O}_n \}$ and $Q(\text{sgn}(A))$ intersect transversally at $A,$ that is, the tangent spaces of ${\cal M}^{\approx}_A $ and $Q(\text{sgn}(A))$ at $A$ sum to $\mathbb R^{m \times n},$ we say that $A$ has the orthogonal equivalence transversality property (OETP). The OETP is instrumental in studying the possible singular values of matrices with a given zero-nonzero pattern. This paper establishes many fundamental properties of matrices with the OETP. For example, it is shown that the set of matrices with the OETP is closed under permutation equivalence, transpose, and multiplication of any rows and columns by $-1$. Further, it is shown by using the Kronecker product that if a direct sum of two matrices $A$ and $B$ has the OETP, then both $A$ and $B$ have the OETP, and $A$ and $B$ do not have any common singular value. A main focus is the identification of classes of zero-nonzero patterns that require the OETP, such as the fully $k$-diagonal patterns with $k\ge 2$. We also introduce the row and column analogs of the OETP, which are stronger and more easily checked properties that may be used for example to determine if a matrix has the OETP by examining certain submatrices. Some problems for further investigation are also posed. This is joint work with M. Arav, H. van der Holst, F. Hall, J. Liu, J. Seo, L. Wang, Y. Xu, and Y. Zhao.
