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 $t$.

中文翻译：

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1 参数矩阵流的合并特征值和交叉特征曲线

我们研究了 1 参数厄米特和一般复或实矩阵流 $A(t)$ 的特征值曲线，根据它们的几何形状和所有参数 $t$ 的 $A(t)$ 的统一可分解性。Hund 和 von Neumann 以及 Wigner 对 1920 年代后期的特征曲线交叉经常错误引用和误用的结果在厄米矩阵流 $A(t) = (A(t))^*$ 中得到了澄清。将这些结果扩展到一般非正态或非厄米特 1 参数矩阵流的猜想被制定和研究。描述并测试了一种计算均匀可分解厄米特矩阵流的块维数的算法。该算法使用 ZNN 方法计算 $A(t)$ 的时变矩阵特征值曲线，用于 $t_o \leq t\leq t_f$。描述了一般复杂矩阵流的类似工作。这种扩展会导致许多新的开放性问题。具体而言，我们指出了一般复杂矩阵流 $A(t)$ 的特征曲线几何形状与一般流通过所有参数 $t$ 的一个固定幺正或一般矩阵相似性可分解为块对角形式之间的困难关系。