论文标题
1-参数基质流的合并特征值和交叉特征库
Coalescing Eigenvalues and Crossing Eigencurves of 1-Parameter Matrix Flows
论文作者
论文摘要
我们根据其几何形状和所有参数$ t $的$ a（t）$的均匀分解性，研究了1参数冬宫和一般复合物或真矩阵的特征值曲线。 Hund和von Neumann经常被误引用和误用的结果以及1920年代后期的特征库杂交的Wigner，均为Hermitean Matrix澄清了$ a（t）=（a（t））^*$。将这些结果扩展到一般的非正常或非弱者1参数基质流动的猜想是制定和研究的。描述并测试了一种计算均匀分解的冬尔米氏矩阵流的块尺寸的算法。该算法使用ZNN方法计算$ a（t）$的时变矩阵特征曲线，用于$ t_o \ leq t \ leq t \ leq t_f $。描述了一般复杂矩阵流的类似努力。此扩展会导致许多新的和开放的问题。具体而言，我们指出了通用复杂矩阵的特征库的几何形状$ a（t）$与一般流量通过所有参数$ t $的固定统一或一般矩阵相似性之间的分解性。
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 $t$.
