对两个极小谱任意符号模式的刻画

Characterization of two minimally spectrally arbitrary sign patterns

一个n阶符号模式矩阵A称为谱任意的,若对给定的任意n次首一实系数多项式f(x),都存在一个实矩阵B∈Q(A),使得B的特征多项式为f(x)。如果谱任意符号模式A的任意一个真子模式都不是谱任意的,则称A为极小谱任意符号模式。给出了两个新的符号模式,运用幂零-雅可比与幂零-中心化两种不同的方法,证明其为极小谱任意符号模式,对两种证明方法进行了比较。

An n × n sign pattern A is called a spectrally arbitrary pattern if for any given real monic polynomial f（x） of degree n, there exists a real matrix B ∈ Q （A） such that the characteristic polynomial of B is f（x）. A sign pattern A is minimally arbitrary if A is a spectrally arbitrary pattern and no proper subpattern of A is spectrally arbitrary. It is proven that two new sign patterns are minimally spectrally arbitrary patterns by using the Nilpotent-Jacobian Method and the Nilpotent-Centralizer method. Furthermore, a comparision of the both methods was also considered.