报告题目 (Title)：$4 \times 4$ Irreducible sign pattern matrices that require four distinct eigenvalues (强迫四个不同特征值的不可约4×4符号模式矩阵)

报告人 (Speaker)： Zhongshan Li 教授（佐治亚州立大学）

报告时间 (Time)：2024年6 月13日(周四) 10:00

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：A sign pattern matrix is a matrix whose entries are from the set $\{+,-, 0\}$. For a sign pattern matrix $A$, the qualitative class of $A$, denoted $Q(A)$, is the set of all real matrices whose entries have signs given by the corresponding entries of $A$. An $n\times n$ sign pattern matrix $A$ requires all distinct eigenvalues if every real matrix in $Q(A)$ has $n$ distinct eigenvalues. In the article ``Sign patterns that require all distinct eigenvalues'', JP J. Algebra Number Theory Appl., 2:2 (2002), 161--179, Li and Harris characterized the $2 \times 2$ and $3\times 3 $ irreducible sign pattern matrices that require all distinct eigenvalues, and established some useful general results on $n\times n$ sign patterns that require all distinct eigenvalues. In this talk, we characterize $ 4\times 4 $ irreducible sign patterns that require four distinct eigenvalues. This is done by characterizing $ 4\times 4 $ irreducible sign patterns that require four distinct real eigenvalues, that require four distinct nonreal real eigenvalues, or that require two distinct real eigenvalues and a pair of conjugate nonreal eigenvalues. The last case turns out to be much more involved. Some interesting open problems are presented. Three important tools that are used in the paper are the following: the discriminant of a polynomial; the fact that if a square sign pattern matrix $A$ requires all distinct eigenvalues then $A$ requires a fixed number of real eigenvalues; and the known result that if $A$ is a ``$k$-cycle'' sign pattern then for each $B \in Q(A)$, the $k$ nonzero eigenvalues of $B$ are evenly distributed on a circle in the complex plane centered at the origin.