符号模式集FSP(3,n-3)中蕴含幂零的模式类

The Potentially Nilpotent Sign Patterns in the Set of Sign Patterns FSP(3,n-3)

以符号集合{+,-,0}中的元素构成的矩阵A称为符号模式矩阵或符号模式.如果对于符号模式A,实矩阵A中的元素与A中对应元素的符号相同,则称A是A的一个实现.如果A有一个实现是幂零矩阵,则称A为蕴含幂零的符号模式.该文引入了一类符号模式矩阵,记为FSP(3,n-3).得出FSP(3,n-3)中所有偶数阶的模式都不是蕴含幂零的,并且给出了n=7阶这种形式的符号模式蕴含幂零的充要条件.

A matrix A whose entries come from the set {＋,-,0} is called a sign pattern matrix, or sign pattern. If A is a sign pattern and A is a real matrix for which each entry has the same sign as the corresponding entry of A, then A is said to be a realization of A. A sign pattern is said to be potentially nilpotent if it has a nilpotent realization. In this paper, a class of sign patterns, denoted by FSP（3, n - 3）, is introduced. The authors determine all potentially nilpotent sign patterns in FSP（3, 4）, and prove that no sign pattern of even order in FSP（3, n - 3） is Dotentially nilDotent.