| 尤利华,邵嘉裕.关于极大$S^2NS$阵的一个注记[J].数学研究及应用,2007,27(1):113~122 |
| 关于极大$S^2NS$阵的一个注记 |
| A Note on the Numbers of Nonzero Entries of Maximal $S^2NS$ Matrices |
| 投稿时间：2005-11-02 修订日期：2006-01-20 |
| DOI：10.3770/j.issn:1000-341X.2007.01.016 |
| 中文关键词: 符号 极大 $S^2NS$ 矩阵 有向图. |
| 英文关键词:sign maximal $S^2NS$ matrices digraphs. |
| 基金项目:国家自然科学基金(10331020);数学天元基金(10526019);广东省博士科研启动基金(5300084). |
|
| 摘要点击次数: 3286 |
| 全文下载次数: 2134 |
| 中文摘要: |
| 一个实方阵$A$称为是$S^2NS$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$S^2NS$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$S^2NS$阵,则称$A$是极大$S^2NS$阵.设所有$n$阶极大$S^2NS$阵的非零元个数所成之集合为${\cal S}(n)$, $Z_4(n)=\{\frac{1}{2}n(n-1)+4, \cdots, \frac{1}{2}n(n+1)-1\}$,除了$2n+1$到$3n-4$间的一段和$Z_4(n)$外,${\cal S}(n)$得到了完全确定.本文将用图论方法证明$Z_4(n)\cap {\cal S}(n)=\phi$. |
| 英文摘要: |
| 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. |
| 查看全文 查看/发表评论 下载PDF阅读器 |
|
|
|
