Positive eigenvector of nonlinear perturbations of nonsymmetric M-matrix and Newton iterative solution

Positive eigenvector of nonlinear perturbations of nonsymmetric M-matrix and its Newton iterative solution are studied. It is shown that any number greater than the smallest positive eigenvalue of the M-matrix is an eigenvalue of the nonlinear problem and that the corresponding positive eigenvector is unique and the Newton iteration of the positive eigenvector is convergent. Moreover, such positive eigenvectors form a monotone increasing and continuous function of the corresponding eigenvalues.     

Generic existence and uniqueness of positive eigenvalues and eigenvectors

We consider a closed cone of positive operators on an ordered Banach space and prove that a generic element of this cone has a unique positive eigenvalue and a unique (up to a positive multiple) positive eigenvector. Moreover, the normalized iterations of such a generic element converge to its unique eigenvector.     

Inverse iteration for calculating the spectral radius of a non-negative irreducible matrix

Noda established the superlinear convergence of an inverse iteration procedure for calculating the spectral radius and the associated positive eigenvector of a non-negative irreducible matrix. Here a new proof is given, based completely on the underlying order structure. The main tool is Hopf's inequality. It is shown that the convergence is quadratic.     

The approach to consistency in the analytic hierarchy process

In his first book on the Analytic Hierarchy Process, T. L. Saaty left open several mathematical questions about the structure of the set of positive reciprocal matrices. In this paper we consider three of these questions: Given an eigenvector and all matrices which give rise to it, can one go from one of them to any order by making small perturbations in the entries? Given two positive column vectors v and w is there a perturbation which carries the set of all positive reciprocal matrices with principal right eigenvector v to the set of positive reciprocal matrices with principal right eigenvector w? Does the set of positive reciprocal n×n matrices whose left and right principal eigenvectors are reciprocals coincide with the set of consistent matrices for n⩾4?     

Size of nodal domains of the eigenvectors of a graph

Consider an eigenvector of the adjacency matrix of a G(n,p) graph. A nodal domain is a connected component of the set of vertices where this eigenvector has a constant sign. It is known that with high probability, there are exactly two nodal domains for each eigenvector corresponding to a nonleading eigenvalue. We prove that with high probability, the sizes of these nodal domains are approximately equal to each other.     

Embeddings of Riemannian manifolds with finite eigenvector fields of connection Laplacian

We study the problem asking if one can embed manifolds into finite dimensional Euclidean spaces by taking finite number of eigenvector fields of the connection Laplacian. This problem is essential for the dimension reduction problem in manifold learning. In this paper, we provide a positive answer to the problem. Specifically, we use eigenvector fields to construct local coordinate charts with low distortion, and show that the distortion constants depend only on geometric properties of manifolds with metrics in the little Hölder space \(c^{2,\alpha }\). Next, we use the coordinate charts to embed the entire manifold into a finite dimensional Euclidean space. The proof of the results relies on solving the elliptic system and providing estimates for eigenvector fields and the heat kernel and their gradients. We also provide approximation results for eigenvector field under the \(c^{2,\alpha }\) perturbation.     

关于M/M/n排队模型的动态解及稳定性

文章讨论动态 M/M/n排队模型 ,运用算子半群理论证明了该模型动态正解的存在唯一性 .并进一步表明零点是系统的一个本征值 ,相应的本征函数为系统的一个定态正解 ,系统的动态正解强稳定到定态解     

Composite orthogonal projection methods for large matrix eigenproblems

For classical orthogonal projection methods for large matrix eigenproblems, it may be much more difficult for a Ritz vector to converge than for its corresponding Ritz value when the matrix in question is non-Hermitian. To this end, a class of new refined orthogonal projection methods has been proposed. It is proved that in some sense each refined method is a composite of two classical orthogonal projections, in which each refined approximate eigenvector is obtained by realizing a new one of some Hermitian semipositive definite matrix onto the same subspace. Apriori error bounds on the refined approximate eigenvector are established in terms of the sine of acute angle of the normalized eigenvector and the subspace involved. It is shown that the sufficient conditions for convergence of the refined vector and that of the Ritz value are the same, so that the refined methods may be much more efficient than the classical ones.     

自反Banach格上正算子基态的存在性

非负算子基态的存在性,是Perron-Frobenius型定理的核心内容,也是证明算子谱缝隙的主要步骤.本文主要利用了正算子非半紧测度的概念,讨论了一般自反Banach格上正算子基态的存在性,并得到了一个充分条件.更为特殊的,若算子同时满足不可约性,证明了基态特征向量是严格正的.     

弱内向型集压缩映射的正不动点和正特征元

本文建立了弱内向型集压缩映射的范数压缩与扩张型正不动点定理，并获得关于正特征元的一些结果．     

Bemerkung zu einer Arbeit von W. Wetterling über positive Operatoren

Summary Recently Wetterling proved several inclusion theorems for the positive eigenvector of a positive operator (i.e. a linear operator mapping a cone in a Banach space into itself). These results can be interpreted in terms of oscillations of vector pairs and thus fit into the general theory of the oscillation of positive operators. The new derivation leads to slightly sharper estimates.     

实H空间对称线性单调算子固有值问题及应用

本文在实Ｈｉｌｂｅｒｔ空间中讨论了带权算子的对称线性单调算子的固有值问题，得出了最小正固有值及其相应的固有向量的一些相关结论，并给出了它们在某些边值、初边值问题等方面的应用     

A simple extrapolation method for clustered eigenvalues

Numerical Algorithms - This paper introduces a simple variant of the power method. It is shown analytically and numerically to accelerate convergence to the dominant eigenvalue/eigenvector pair,...     

On polyhedra of Perron-Frobenius eigenvectors

This paper deals with the positive eigenvectors of nonnegative irreducible matrices which are merely characterized by a given upper bound u on their spectral radius and by a given matrix L of lower bounds for their elements. For any such matrix, the normalized positive left [right] eigenvector is shown to belong to the polyhedron the vertices of which are given by the normalized rows [columns] of the matrix $\text{(u}\text{I}\text{?}\text{L}\text{)}{}^{\mathrm{?1}}$. This polyhedron is proven to be also the smallest closed set which is guaranteed to contain the positive left [right] normalized eigenvector; its vertices are therefore the best componentwise bounds one can obtain on the positive eigenvectors of these matrices. A less general result has also been obtained for the symmetrical case, when the matrices are only characterized by a given lower bound l on their spectral radius and by a given matrix U of upper bounds for their elements.     

Characterization of symmetric M-matrices as resistive inverses

We aim here at characterizing those nonnegative matrices whose inverse is an irreducible Stieltjes matrix. Specifically, we prove that any irreducible Stieltjes matrix is a resistive inverse. To do this we consider the network defined by the off-diagonal entries of the matrix and we identify the matrix with a positive definite Schrödinger operator whose ground state is determined by the lowest eigenvalue of the matrix and the corresponding positive eigenvector. We also analyze the case in which the operator is positive semidefinite which corresponds to the study of singular irreducible symmetric M-matrices.     

Composite orthogonal projection methods for large matrix eigenproblems

For classical orthogonal projection methods for large matrix eigenproblems, it may be much more difficult for a Ritz vector to converge than for its corresponding Ritz value when the matrix in question is non-Hermitian. To this end, a class of new refined orthogonal projection methods has been proposed. It is proved that in some sense each refined method is a composite of two classical orthogonal projections, in which each refined approximate eigenvector is obtained by realizing a new one of some Hermitian semipositive definite matrix onto the same subspace. Apriori error bounds on the refined approximate eigenvector are established in terms of the sine of acute angle of the normalized eigenvector and the subspace involved. It is shown that the sufficient conditions for convergence of the refined vector and that of the Ritz value are the same, so that the refined methods may be much more efficient than the classical ones. Project supported by the China State Major Key Projects for Basic Researches, the National Natural Science Foundation of China (Grant No. 19571014), the Doctoral Program (97014113), the Foundation of Excellent Young Scholors of Ministry of Education, the Foundation of Returned Scholars of China and the Liaoning Province Natural Science Foundation.     

经济最优化意义下消耗系数的调整

本文基于“正特片矢量法”,提出了一种在经济最优化意义下调整消耗系数的方法.该方法能够综合考虑产综和价格对消耗系数的影响,有着较好的普适性,在计算上简便易行、效果良好,且具有明显的经济意义.     

Properties of a matrix with a very ill-conditioned eigenproblem

It is proved that a matrix that has a very ill-conditioned eigenvector matrix is close to one that has multiple eigenvalues, and an estimate of this distance is given, measured in the Euclidean matrix norm.     

A new differential equation method for finding the Perron root of a positive matrix

A basic problem in linear algebra is the determination of the largest eigenvalue (Perron root) of a positive matrix. In the present paper a new differential equation method for finding the Perron root is given. The method utilizes the initial value differential system developed in a companion paper for individually tracking the eigenvalue and corresponding right eigenvector of a parametrized matrix.     

On Spectral Hypergraph Theory of the Adjacency Tensor

We study both H and E/Z-eigenvalues of the adjacency tensor of a uniform multi-hypergraph and give conditions for which the largest positive H or Z-eigenvalue corresponds to a strictly positive eigenvector. We also investigate when the E-spectrum of the adjacency tensor is symmetric.