关于极大$S^2NS$阵的一个注记

一个实方阵A称为是S2NS阵,若所有与A有相同符号模式的矩阵均可逆,且它们的逆矩阵的符号模式都相同．若A是S2NS阵且A中任意一个零元换为任意非零元后所得的矩阵都不是S2NS阵,则称A是极大S2NS阵．设所有n阶极大S2NS阵的非零元个数所成之集合为S(n),Z4(n)={1/2n(n-1) 4,…,1/2n(n 1)-1},除了2n 1到3n一4间的一段和Z4(n)外,S(n)得到了完全确定．本文将用图论方法证明Z4(n)∩S(n)=(?)．

A Note on the Numbers of Nonzero Entries of Maximal $S^2NS$ Matrices

A square real matrix $A$ is called an $S^2NS$ matrix, if every matrix with the same sign pattern as $A$ is invertible, and the inverses of all such matrices have the same sign pattern. A matrix $A$ is called a maximal $S^2NS$ matrix, if $A$ is an $S^2NS$ matrix, but each matrix obtained from $A$ by replacing one zero entry by a nonzero entry is not a $S^2NS$ matrix. Let ${\cal S}(n)$ be the set of numbers of nonzero entries of maximal $S^2NS$ matrices with order $n~(\geq 5),$ and $Z_4(n)=\{\frac{1}{2}n(n-1)+4, \cdots, \frac{1}{2}n(n+1)-1\}$. We know that ${\cal S}(n)$ has been described except for the numbers between $2n+1$ and $3n-4$ and the numbers in $Z_4(n)$. We prove $Z_4(n)\cap {\cal S}(n)=\phi$ by graphic method in this paper.