单参数矩阵流的特征值和交叉特征曲线的合并
Coalescing Eigenvalues and Crossing Eigencurves of 1-Parameter Matrix Flows
摘要：
本文研究了单参数厄米矩阵流和一般复矩阵流或实矩阵流$A(t)$的特征值曲线，并讨论了$A(t)$对所有参数$t$的一致可分解性. Hund和von Neumann以及Wigner在20世纪20年代后期关于本征曲线交叉的经常被错误引用和误用的结果，在$A(t) = (A(t))^*$的厄米矩阵流中得到了澄清。将这些结果推广到一般非正规或非厄米单参数矩阵流的猜想制定和调查。描述并测试了一种计算可均匀分解的厄米特矩阵流的块维数的算法。该算法使用ZNN方法计算$t_o \leq t\leq t_f$的$A(t)$的时变矩阵特征值曲线。类似的努力一般复杂的矩阵流。这一延伸导致了许多新的和开放的问题。具体来说，我们指出的困难之间的关系的几何特征曲线一般复杂的矩阵流$A(t)$和一般的流的分解成块对角形式通过一个固定的酉或一般矩阵相似性的所有参数$t$。
英文摘要：
We investigate the eigenvalue curves of 1-parameter hermitean and general complex or real matrix flows $A(t)$ in light of their geometry and the uniform decomposability of $A(t)$ for all parameters $t$. The often misquoted and misapplied results by Hund and von Neumann and by Wigner for eigencurve crossings from the late 1920s are clarified for hermitean matrix flows $A(t) = (A(t))^*$. A conjecture on extending these results to general non-normal or non-hermitean 1-parameter matrix flows is formulated and investigated. An algorithm to compute the block dimensions of uniformly decomposable hermitean matrix flows is described and tested. The algorithm uses the ZNN method to compute the time-varying matrix eigenvalue curves of $A(t)$ for $t_o \leq t\leq t_f$. Similar efforts for general complex matrix flows are described. This extension leads to many new and open problems. Specifically, we point to the difficult relationship between the geometry of eigencurves for general complex matrix flows $A(t)$ and a general flow's decomposability into blockdiagonal form via one fixed unitary or general matrix similarity for all parameters $t$.
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