Similarity via transversal intersection of manifolds


3:30 PM - 4:30 PM



  • Time:15:30-16:30 pm (Beijing Time)
  • Date: Friday, 14th June, 2024
  • Venue: MB441
  • Speaker:Prof Zhongshan (Jason) Li, Georgia State University
  • Language: English


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.


Professor Zhongshan Li was born in Lanzhou, China, and is currently a tenured full professor in the Department of Mathematics at Georgia State University in the United States. His research interests include combinatorial matrix theory, algebraic graph theory, applications of matrix theory, and more.
Professor Li graduated with a Bachelor of Science in Mathematics from Lanzhou University in 1983, obtained a Master of Science in Mathematics from Beijing Normal University in 1986, and earned a Ph.D. in Mathematics from North Carolina State University in 1990. Since 1991, he has been teaching in the Department of Mathematics and Statistics at Georgia State University in the United States, became an associate professor and tenured professor at Georgia State University in 1998, and was promoted to full professor in 2007. Since 2010, he has served as the Director of Graduate Studies in the Department of Mathematics and became a member of the Promotion and Tenure Committee of the College of Arts and Sciences at Georgia State University in 2010.
Professor Li has been invited to attend and present papers at numerous international mathematical conferences and has delivered academic reports at dozens of institutions, including Peking University, the Academy of Mathematics and Systems Science of the Chinese Academy of Sciences, Tsinghua University, Beijing Normal University, Nankai University, Fudan University, Tongji University, University of Science and Technology of China, Emory University, University of Wisconsin, Auburn University, University of Tennessee, Shanghai Jiao Tong University, East China Normal University, Shanghai University, Huazhong Normal University, Lanzhou University, Shandong University, Ocean University of China, North University of China, University of Electronic Science and Technology of China, Fuzhou University, Harbin Engineering University, Heilongjiang University, Changsha University of Science and Technology, Xiangtan University, Northwest Normal University, and others. He has published over 80 papers in prestigious international academic journals such as the "American Mathematical Monthly," "Linear Algebra and Its Applications," "SIAM J. on Discrete Mathematics," "J. Combin. Theory Ser. B," "Linear and Multilinear Algebra," "Graphs and Combinatorics," "IEEE Transactions on Neural Networks and Learning Systems," and has written a chapter on sign pattern matrices for the academic monograph "Handbook of Linear Algebra." He has led or participated in numerous research projects. His current research mainly focuses on combinatorial matrix theory, including sign pattern matrices, minimum rank problems, eigenvalue problems, matrix manifolds, algebraic graph theory, integer matrices, and sign vector sets of real linear subspaces.
Professor Li also serves as a featured reviewer for "Mathematical Reviews" in the United States and is on the editorial boards of the "JP Journal of Algebra, Number Theory and Applications" and "Special Matrices" journals. He has served as a project review expert for the Natural Sciences and Engineering Research Council of Canada in 2008-2009, 2015-2016, and 2018-2019.
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