惯量任意的符号模式

矩阵A的特征值的集合(含重数)记为σ(A),A的惯量是指三元有序数组i(A)=(i (A),i-(A),i0(A)),其中i (A),i-(A)和i0(A)分别表示具有正,负,零实部特征值的个数.n阶符号模式矩阵S=(sij)是指元素取自{1,-1,0}或者{ ,-,0}的矩阵,S的定性矩阵类是指集合Q(S)={A=(aij)∈Mn(R):对所有的i和j,sign(aij)=sij}.S的惯量是指集合i(S)={i(A):A∈Q(S)}.若对任意满足n1 n2 n3=n的非负三元数组(n1,n2,n3),都有(n1,n2,n3)∈i(S),则称符号模式S为惯量任意模式.考虑n阶符号模式Kn=(kij)n×n:当1≤j-i≤n-2或i=j=n时,kij=1;当1≤i-j≤n-2或i=j=1时,kij=-1;当|i-j|=n-1时,kij可以取任意固定值;其余情形时,kij=0.本文证明了Kn(n≥3)是惯量任意模式.

Sign Patterns with Arbitrary Inertia

The set of all eigenvalues (counting multiplicities) of a matrix A is denoted by σ(A),and the inertia of A is the ordered triple i(A)=(i+(A),i-(A),i0(A)),in which i+(A),i-(A) and i0(A) are the numbers of eigenvalues with positive,negative and zero real parts,respectively. An n×n sign pattern S=(sij) has sij∈{1,-1,0} or sij∈{+,-,0},and the qualitative class of S is Q(S)={A=(aij)∈Mn(R):sign(aij)=sij for all i,j}. The inertia of S is the set of ordered triples i(S)={i(A):A∈Q(S)}. An n×n sign pattern S is an inertially arbitrary pattern (IAP) if (n1,n2,n3)∈i(S) for each nonnegative triple (n1,n2,n3) with n1+n2+n3=n. Consider the n ×n sign pattern Kn,where Kn is the pattern with positive entry (i,j) for 1≤ j-i≤n-2 or i=j=n,negative entry (i,j) for 1≤ i-j≤n-2 or i=j=1,arbitrary entry (i,j) for |i-j|=n-1 and zero entry otherwise. In this paper,it is proved that Kn is an IAP for n≥3.