两类惯量惟一的对称符号模式

§ 1　 IntroductionA sign pattern(matrix) A is a matrix whose entries are from the set{ +,-,0 } .De-note the setofall n× n sign patterns by Qn.Associated with each A=(aij)∈ Qnis a class ofreal matrices,called the qualitative class of A,defined byQ(A) ={ B =(bij)∈ Mn(R) |sign(bij) =aijfor all i and j} .　　 For a symmetric sign pattern A∈ Qn,by G(A) we mean the undirected graph of A,with vertex set { 1 ,...,n} and (i,j) is an edge if and only if aij≠ 0 .A sign pattern A∈ Qnis a do…

Two classes of symmetric sign patterns that require unique inertia

A sign pattern is a matrix whose entries are from the set {+,−,0}. Associated with each sign pattern A of order n is a qualitative class of A,defined by Q(A). For a symmetric sign pattern A of order n,the inertia of A is a set i(A)={i(B)=(i ₊(B),i ₋(B),i ₀(B)) | B=B ^T H∈Q(A)}, where i ₊ (B) (respectively,i ₋ (B),i ₀(B) denotes the number of positive (respectively, negative, zero) eigenvalues. That the symmetric sign pattern A requires unique intertia means i(B ₁)=i(B ₂) for all real symmetric matrices B ₁,B ₂∈Q(A). The purpose of this paper is to characterize double star and cycle sign patterns that require unique inertia. Further, their unique inertia is also obtained. Supported by Shanxi Natural Science Foundation (20011006).