摘要
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 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 8（n） be the set of numbers of nonzero entries of maximal S^2NS matrices with order n （≥）, and Z4（n） = {1/2（n-1）＋4,…,1/2n（n＋1）-1}. We know that 8（n） has been described except for the numbers between 2n ＋ 1 and 3n- 4 and the numbers in Z4（n）. We prove Z4（n） ∩ 8（n） = φ by graphic method in this paper.
基金
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).
