M-矩阵最小特征值的上界估计

利用M-矩阵最小特征值与非负矩阵谱半径之间的关系,结合矩阵的迹分两种情况给出M-矩阵最小特征值的上界序列,并且给出数值例子加以说明.     

严格对角占优M-矩阵的逆矩阵的无穷大范数的新上界

给出了严格对角占优M-矩阵的逆矩阵的无穷大范数上界新的估计式,进而给出严格对角占优M-矩阵的最小特征值下界的估计式.新估计式改进了已有文献的结果.     

Cassini卵形谱包含域的改进及应用

本文给出了改进的Cassini卵形谱包含域,讨论了相应的隔离定理及边界问 题,所得结果推广了文[1-4]的相应定理.作为应用得到了M-矩阵的一个新表征.     

树的第二个最大特征值

本文得到了边独立数为n且阶为2n+2的树的第二个最大特征值的精确上界,且给出了达到上界的所有的极树.     

四阶不定微分算子非实特征值的界

给出了四阶正则不定微分算子仅在可积条件下的非实特征值上界和下界的估计.更一般地,非实特征值的下界可以利用Krein空间的自共轭算子得到.     

M-矩阵最小特征值下界的新不等式

非奇异M-矩阵B的最小特征值τ(B)的下界是矩阵论中重要的研究课题.利用特征值定位定理,首先给出非负矩阵与M-矩阵的逆矩阵Hadamard积的谱半径上界,进而给出M-矩阵最小特征值下界的新不等式.新不等式只与矩阵的元素有关,易于计算.理论分析和数值例子表明所给结果改进了现有结果.     

高阶张量Pareto-特征值的若干性质

考虑高阶张量特征值互补问题,由于求解张量的最大Pareto-特征值是一个NP难问题,关注于Pareto-特征值的估计,并给出若干关于Z-张量和M-张量的Pareto-特征值的性质.     

矩阵Hadamard积和Fan积的特征值界的一些新估计式（英文）

本文研究了非奇异M-矩阵A与B的Fan积的最小特征值下界和非负矩阵A与B的Hadamard积的谱半径上界的估计问题.利用Brauer定理,得到了一些只依赖于矩阵的元素且易于计算的新估计式,改进了文献[41现有的一些结果.     

严格α-对角占优M-矩阵A的‖A~(-1)‖_∞的上界估计

针对逆矩阵的无穷范数的上界估计问题,利用严格对角占优M-矩阵逆的无穷范数的上界,给出了严格α-对角占优M-矩阵A的||A~(-1)||_∞的单调递减的上界序列,理论证明及数值分析均表明所得估计改进了某些现有结果.     

球面域上高阶拉普拉斯的特征值估计

本文研究了球面域上高阶拉普拉斯的特征值问题. 利用Rayleigh-Ritz不等式, 获得了球面域上高阶拉普拉斯的第(k+1)个特征值的上界估计, 这个估计式由前k个特征值给出.     

Upper bounds for eigenvalues of Cauchy-Hankel tensors

We present upper bounds of eigenvalues for finite and infinite dimensional Cauchy-Hankel tensors. It is proved that an m-order infinite dimensional Cauchy-Hankel tensor defines a bounded and positively (m－1)-homogeneous operator from l¹ into l^p (1<p<∞); and two upper bounds of corresponding positively homogeneous operator norms are given. Moreover, for a fourth-order real partially symmetric Cauchy-Hankel tensor, suffcient and necessary conditions of M-positive definiteness are obtained, and an upper bound of M-eigenvalue is also shown.     

利用谱投影梯度法计算张量M-特征值（英文）

In this paper, we present the connection between the M-eigenvalues of a fourth-order partially symmetric rectangular tensor and the Z-eigenvalues of a new fourth-order weakly symmetric square tensor by using the symmetric embedding technique. Based on this, the M-eigenvalue problem can be converted to be the Z-eigenvalue problem. Then we compute the M-eigenpairs by the spectral projected gradient(SPG) method that is for computing the Z-eigenpairs. Some numerical results are reported at the end of this paper.     

-inclusion sets for eigenvalues of a tensor

In this paper, we mainly focus on new inclusion sets for eigenvalues of a tensor. First, we propose new inclusion sets for eigenvalues of a tensor, which are sharper than some existing inclusion sets, and obtain the law of distribution of the number of eigenvalues for a tensor. Second, two new classes of tensors are introduced. Third, some bounds on the spectral radii for nonnegative tensors are given. Fourth, some checkable sufficient conditions for the positive definiteness (positive semidefiniteness) of some classes of even-order real symmetric tensors are obtained.     

New eigenvalue inclusion sets for tensors

Two new eigenvalue inclusion sets for tensors are established. It is proved that the new eigenvalue inclusion sets are tighter than that in Qi's paper “Eigenvalues of a real supersymmetric tensor”. As applications, upper bounds for the spectral radius of a nonnegative tensor are obtained, and it is proved that these upper bounds are sharper than that in Yang's paper “Further results for Perron–Frobenius theorem for nonnegative tensors”. And some sufficient conditions of the positive definiteness for an even‐order real supersymmetric tensor are given. Copyright © 2012 John Wiley & Sons, Ltd.     

Singular value inclusion sets for rectangular tensors

Two singular value inclusion sets for rectangular tensors are given. These sets provide two upper bounds and lower bounds for the largest singular value of nonnegative rectangular tensors, which can be taken as a parameter of an algorithm presented by Zhou et al. (Linear Algebra Appl. 2013; 438: 959–968) such that the sequences produced by this algorithm converge rapidly to the largest singular value of an irreducible nonnegative rectangular tensor.     

关于非负张量Z-谱半径的一些结果（英文）

Z-eigenvalue plays a fundamental role in the best rank-one approximation.Chang,Pearson,and Zhang generalized the Perron-Probenius theorem for Z-eigenvalues of nonnegative tensors and gave some properties of the Z-spectral radius recently.In this paper,we give some properties of Z-eigenvectors associated with Z-spectral radius of nonnegative weakly symmetric tensors,compare the Zspectral radius between two nonnegative tensors,and modify the upper and lower bounds for Z-spectral radius.Some results for the Z-singular values of rectangular tensors are also given.     

Bounds for the Perron root, singularity/nonsingularity conditions, and eigenvalue inclusion sets

Using a unified approach based on the monotonicity property of the Perron root and its circuit extension, a series of exact two-sided bounds for the Perron root of a nonnegative matrix in terms of paths in the associated directed graph is obtained. A method for deriving the so-called mixed upper bounds is suggested. Based on the upper bounds for the Perron root, new diagonal dominance type conditions for matrices are introduced. The singularity/nonsingularity problem for matrices satisfying such conditions is analyzed, and the associated eigenvalue inclusion sets are presented. In particular, a bridge connecting Gerschgorin disks with Brualdi eigenvalue inclusion sets is found. Extensions to matrices partitioned into blocks are proposed.     

Improved Hashin–Shtrikman Bounds for Elastic Moment Tensors and an Application

This paper is devoted to the derivation of trace bounds for elastic moment tensors. Starting from the integral equation formulation of the elastic moment tensor, we establish that its trace can be obtained as a sum of minimal energies. We then recover the so-called Hashin–Shtrikman bounds, and show that these bounds can be tightened for inclusions which have some local thickness. As an application, we show that the volume of the inclusion can be estimated by the elastic moment tensor. Y.C. is partially supported by the grants RTN MULTIMAT and ANR EchoScan. H.K. is partially supported by the grant KOSEF R01-2006-000-10002-0.     

Structural Properties of Tensors and Complementarity Problems

In this paper, one of our main purposes is to prove the boundedness of the solution set of tensor complementarity problems such that the specific bounds depend only on the structural properties of such a tensor. To achieve this purpose, firstly, we prove that this class of structured tensors is strictly semi-positive. Subsequently, the strictly lower and upper bounds of operator norms are given for two positively homogeneous operators. Finally, with the help of the above upper bounds, we show that the solution set of tensor complementarity problems has the strictly lower bound. Furthermore, the upper bounds of spectral radius are obtained, which depends only on the principal diagonal entries of tensors.