几类特殊的谱任意和惯量任意符号模式矩阵

Some Spectrally and Inertially Arbitrary Sign Patterns

利用矩阵理论和组合论的方法对一些特殊类型的符号模式的惯量和谱进行了研究.找到了一类谱任意的零-非零符号模式矩阵,同时也是惯量任意的符号模式矩阵,证明了Hessenberg型矩阵对应的符号模式矩阵是极小惯量任意符号模式矩阵;给出一类由低阶矩阵通过直和构造而成的几乎完全惯量任意符号模式矩阵;通过对形如"S"的特殊符号模式矩阵的分析,利用矩阵分解的性质得到了谱任意符号模式矩阵的一个必要条件.

By combinatorial and matrix theoretical methods, the study is focused on some special sign pattern matrices. A class of spectrally and inertially arbitrary zero-nonzero pattern is given. It is proved that the Hessenberg-matrix is the minimal inertially arbitrary sign pattern. An almost inertially arbitrary sign pattern that is constructed by the direct sum from those of lower order is introduced. Based on the property of matrix-decomposition for analyzing S-shaped sign pattern matrix, a necessary condition on the spectrally arbitrary sign pattern matrix is characterized.