两个谱任意模式

设λ1,λ2,...,λn(可以相同)为实矩阵A的所有特征值,记为σ(A)=(λ1,λ2,...,λn).n阶符号模式矩阵S=(sij)是指元素取自{ ,-,0}的矩阵,S的定性矩阵类是指集合Q(S)={A=(aij)∈M\{n\}(R):对所有的i和j,sign(aij)=sij},记σ(S)={σ(A):A∈Q(S)}.设S为n阶符号模式矩阵,λ1,λ2,…,λn为n个任意复数,若λ1,λ2,…,λn中的虚数都与其共轭复数成对出现时,便存在A∈Q(S),使得σ(A)=(λ1,λ2,…,λn),则称S为谱任意模式.在本文中,我们得到两个谱任意模式.

Two Spectrally Arbitrary Patterns

Let λ1,λ2,...,λn(not necessarily distinct) be the all eigenvalues of a real matrix A and σ(A)=(λ1,λ2,...,λn).An n×n sign pattern S=(sij) has sij∈{+,-,0} and its qualitative class of S is Q(S)={A=(aij)∈Mn(R):sign(aij)=sij for all i,j }.For a sign pattern S,define that σ(S)={σ(A):A∈Q(S)}.An n×n sign pattern S is called the spectrally arbitrary pattern provided that for any complex numbers λ1 ,λ2,...,λn with nonreals occurring as conjugate pairs,there exists a matrix A∈Q(S) such that σ(A)=(λ1,λ2,...,λn).In this paper,two spectrally arbitrary patterns are obtained.