矩阵特征值在椭圆形区域上的估计

On localization of eigenvalues in elliptic region

矩阵特征值的估计在理论和应用上都非常重要,传统估计的结果都是用圆形、卵形等区域来定位的。本文寻求一种新的方法来定位复矩阵特征值的分布范围,给出了任意n阶具有实系数特征多项式的矩阵特征值都包含在下面的椭圆形区域内β2(x-trA/n)2+α2y2≤α2β2,其中α=[(n-1)/n∑nk=1(Reλk-trA/n](1/2),β=[(n-1)/n∑nk=1(Imλk)2](1/2)。最后给出了更精确的估计区域,进一步改进了已有的一些结论。

The estimation and location of eigenvalues of complex matrix are very important, and the traditional estimation results are round, oval and other regions. Some new methods are discussed to locate complex matrices＇ eigenvalues. It is shown that all eigenvalues of arbitrarily given n × n complex matrix with its characteristic polynomial having real coefficients can be located by the following elliptic regionβ2（x-trA/n）2＋α2y2≤α2β2,where α=[n-1/n n∑k=1（Reλk-trA/n）2]1/2 and β=[n-1/n n∑k=1（Imλk）2]1/2.Finally, some more precise estimation areas to further improve previous conclusions are obtained.