On the Spectra of Striped Sign Patterns

Sign patterns consisting of some positive and some negative columns, with at least one of each kind, are shown to allow any self-conjugate spectrum, and thus to allow any inertia. In the case of the n × n sign pattern with all columns positive, given any self-conjugate multiset consisting of n m 1 complex numbers supplemented by a sufficiently large positive number, it is shown how to construct a positive normal matrix whose spectrum is this multiset. Thus, the positive sign pattern allows any inertia with at least one positive eigenvalue.     

INERTIA SETS OF SYMMETRIC SIGN PATTERN MATRICES

1　IntroductionIn qualitative and combinatorial matrix theory,we study properties ofa matrix basedon combinatorial information,such as the signs of entries in the matrix.A matrix whoseentries are from the set{ + ,-,0 } is called a sign pattern matrix ( or sign pattern,or pat-tern) .We denote the setof all n× n sign pattern matrices by Qn.For a real matrix B,sgn( B) is the sign pattern matrix obtained by replacing each positive( respectively,negative,zero) entry of B by+ ( respectively,-,0 )…     

带导数项的半正Right Focal边值问题单调正解的存在性

研究下列半正Right F0cal边值问题单调正解的存在性其中λ>0是一个参数,n≥3,1相似文献   

New results about semi-positive matrices

Our purpose is to present a number of new facts about the structure of semipositive matrices, involving patterns, spectra and Jordon form, sums and products, and matrix equivalence, etc. Techniques used to obtain the results may be of independent interest. Examples include: any matrix with at least two columns is a sum, and any matrix with at least two rows, a product, of semipositive matrices. Any spectrum of a real matrix with at least 2 elements is the spectrum of a square semipositive matrix, and any real matrix, except for a negative scalar matrix, is similar to a semipositive matrix. M-matrices are generalized to the non-square case and sign patterns that require semipositivity are characterized.     

Computing the inertia from sign patterns

A symmetric matrix A is said to be sign-nonsingular if every symmetric matrix with the same sign pattern as A is nonsingular. Hall, Li and Wang showed that the inertia of a sign-nonsingular symmetric matrix is determined uniquely by its sign pattern. The purpose of this paper is to present an efficient algorithm for computing the inertia of such symmetric matrices. The algorithm runs in time for a symmetric matrix of order n with m nonzero entries. In addition, it is shown to be NP-complete to decide whether the inertia of a given symmetric matrix is not determined by its sign pattern.     

Sign patterns for eigenmatrices of nonnegative matrices

For a square (0,?1,??1) sign pattern matrix S, denote the qualitative class of S by Q(S). In this article, we investigate the relationship between sign patterns and matrices that diagonalize an irreducible nonnegative matrix. We explicitly describe the sign patterns S such that every matrix in Q(S) diagonalizes some irreducible nonnegative matrix. Further, we characterize the sign patterns S such that some member of Q(S) diagonalizes an irreducible nonnegative matrix. Finally, we provide necessary and sufficient conditions for a multiset of real numbers to be realized as the spectrum of an irreducible nonnegative matrix M that is diagonalized by a matrix in the qualitative class of some S ² NS sign pattern.     

Sign patterns allowing nilpotence of index 3

Suppose P is a property referring to a real matrix. We say that a sign pattern A allows P if there exists at least one matrix with the same sign pattern as A that has the property P. In this paper, we study sign patterns allowing nilpotence of index 3. Four methods for constructing sign patterns that allow nilpotence of index 3 are obtained. All tree sign patterns that allow nilpotence of index 3 are characterized. Sign patterns of order 3 that allow nilpotence are identified.     

The Metapositive Definite Self－Conjugate Solution of the Matrix Equation AXB＝C over a Skew Field

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立方非负的不可约符号模式

以 ，-，0为元素的矩阵称为符号模式矩阵．本文考虑符号模式矩阵的立方模式，给出了至少含一个负元的不可约符号模式的立方模式是非负的充要条件及在置换相似下的标准型；得到了立方非负的符号模式矩阵中出现的正元素的个数的最大(小)值及给出了达到这些最大(小)值的不可约符号模式矩阵的完整刻画．     

Convex central configurations for the n-body problem

We give a simple proof of a classical result of MacMillan and Bartky (Trans. Amer. Math. Soc. 34 (1932) 838) which states that, for any four positive masses and any assigned order, there is a convex planar central configuration. Moreover, we show that the central configurations we find correspond to local minima of the potential function with fixed moment of inertia. This allows us to show that there are at least six local minimum central configurations for the planar four-body problem. We also show that for any assigned order of five masses, there is at least one convex spatial central configuration of local minimum type. Our method also applies to some other cases.     

一类新的极小谱任意符号模式

若给定任意一个$n$次首一实系数多项式$f(\lambda)$,都存在一个实矩阵$B\in Q(A)$, 使得$B$的特征多项式为$f(\lambda)$,则称$A$为谱任意符号模式. 如果一个谱任意符号模式的任意非零元被零取代后所得到的符号模式不是谱任意,那么这个谱任意符号模式称为极小谱任意符号模式.本文证明一类极小谱任意符号模式.     

Sign patterns that allow diagonalizability revisited

Characterization of sign patterns that allow diagonalizability has been a long-standing open problem. In this article, necessary and sufficient conditions for a sign pattern to allow diagonalizability are obtained, in terms of allowing related properties. Some properties of normal sign patterns are considered. In particular, it is shown that normal sign patterns of order up to 3 allow diagonalizability. Two combinatorial necessary conditions for a sign pattern to allow diagonalizability are also presented.     

The Inertia Set of Nonnegative Symmetric Sign Pattern with Zero Diagonal

The inertia set of a symmetric sign pattern A is the set i(A) = {i(B) | B = B ^T ∈ Q(A)}, where i(B) denotes the inertia of real symmetric matrix B, and Q(A) denotes the sign pattern class of A. In this paper, a complete characterization on the inertia set of the nonnegative symmetric sign pattern A in which each diagonal entry is zero and all off-diagonal entries are positive is obtained. Further, we also consider the bound for the numbers of nonzero entries in the nonnegative symmetric sign patterns A with zero diagonal that require unique inertia.     

Inertially arbitrary patterns

An n×n sign pattern matrix A is an inertially arbitrary pattern (IAP) if each non-negative triple (rst) with r+s+t=n is the inertia of a matrix with sign pattern A. This paper considers the n×n(n≥2) skew-symmetric sign pattern S_n with each upper off-diagonal entry positive, the (1,1) entry negative, the (nn) entry positive, and every other diagonal entry zero. We prove that S_n is an IAP.     

On the inertia sets of some symmetric sign patterns

A matrix whose entries consist of elements from the set {+, −, 0} is a sign pattern matrix. Using a linear algebra theoretical approach we generalize of some recent results due to Hall, Li and others involving the inertia of symmetric tridiagonal sign matrices.     

加强P除环上自共轭矩阵的几个定理

证明了如果Ｏ是加强Ｐ－除环,则为实封闭域．利用该结果还讨论了加强Ｐ一除环上自共轭矩阵的正定性．     

On the pebbling threshold of paths and the pebbling threshold spectrum

A configuration of pebbles on the vertices of a graph is solvable if one can place a pebble on any given root vertex via a sequence of pebbling steps. A function is a pebbling threshold for a sequence of graphs if a randomly chosen configuration of asymptotically more pebbles is almost surely solvable, while one of asymptotically fewer pebbles is almost surely not. In this paper we tighten the gap between the upper and lower bounds for the pebbling threshold for the sequence of paths in the multiset model. We also find the pebbling threshold for the sequence of paths in the binomial model. Finally, we show that the spectrum of pebbling thresholds for graph sequences in the multiset model spans the entire range from n^1/2 to n, answering a question of Czygrinow, Eaton, Hurlbert and Kayll. What the spectrum looks like above n remains unknown.     

Symmetric sign patterns with maximal inertias

The inertia of an n by n symmetric sign pattern is called maximal when it is not a proper subset of the inertia of another symmetric sign pattern of order n. In this note we classify all the maximal inertias for symmetric sign patterns of order n, and identify symmetric sign patterns with maximal inertias by using a rank-one perturbation.     

Gaps in the base set of primitive nonpowerful sign patterns

For a primitive nonpowerful square sign pattern A, the base of A, denoted by l(A), is the least positive integer l such that every entry of A ^l is #. In this article, we consider the base set of the primitive nonpowerful sign pattern matrices. Some useful results about the bases for the sign pattern matrices are presented there. Some special sign pattern matrices with given bases are characterized and more ‘gaps’ in the base set are shown.     

一个加性混合幂丢番图不等式Ⅲ

证明了:如果λ1,…,λ11,μ是非零实数,并且不同一符号,至少有一个λi/λj是无理数,那么对任意实数η和ε>0,不等式λ1x14 … λ11x141 μy2 η<ε有无穷多正整数解x1,…,x11,y.