蕴含幂零性与谱任意符号模式矩阵

蕴含幂零性与谱任意符号模式矩阵是近几年组合数学中比较热门的一个研究方向。本文主要运用幂零一中心化子方法来证明谱任意。首先揭示了蕴含幂零及幂零指数与谱任意之间的关系，即：蕴含幂零的符号模式矩阵是谱任意的必要非充分条件且幂零指数为n的符号模式矩阵既非谱任意的必要条件也非充分条件。然后通过几类低阶的幂零矩阵构造了几类高阶的蕴含幂零符号模式矩阵和谱任意符号模式矩阵。最后给出了谱任意符号模式矩阵的直和仍为谱任意符号模式矩阵的一个新的条件。本文对构造幂零矩阵与谱任意符号模式矩阵有一定的应用价值。

Potential Nilpotent and Spectrally Arbitrary Patterns

Potential nilpotent and the spectrally arbitrary sign pattern matrices are a popular research opinion recently. The nilpotent- centralizer method was our main tool to prove spectrally arbitrary throughout our paper. Firstly, It revealed that the potential nilpo- tent sign pattern matrices are necessary but not sufficient condition for spectrally arbitrary patterns. Meanwhile, the sign pattern matrices whose nilpotent index is n have no necessary and sufficient conditions with the spectrally arbitrary patterns. Then, we con- structed some classes of upper order potential nilpotent sign pattern matrices and spectrally arbitrary sign pattern matrices via lower order spectrally arbitrary pattens. Finally, a new condition for the direct sum of some spectrally arbitrary sign pattern matrices to be still spectrally arbitrary was also presented. The paper has important applications in constructing potential nilpotent sign pattern matrices and spectrally arbitrary sign pattern matrices.