Bounds and perturbation bounds for the matrix exponential

Some new types of bounds and perturbation bounds, based on the Jordan normal form, for the matrix exponential are derived. These bounds are compared to known bounds, both theoretically and by numerical examples. Some recent results on the matrix exponential and the logarithmic norm are also included.     

高次矩阵方程f（X）=0的一种解法

关于m次矩阵方程Xm+a1Xm-1+…+am-1X+amEn=0,其中En是n阶单位矩阵,a1,a2,…,am∈R,X∈Cn×n,本文利用矩阵的化零多项式,最小多项式的相关结论以及Jordan标准形分解,讨论了该方程的所有可能解.     

矩阵函数的性质

研究矩阵函数的若干性质,推广了文献[1]的部分结果.     

A short proof of the jordan decomposition theorem

We present a short proof of the Jordan Decomposition Theorem     

The Dimensions of Certain Symmetry Classes of Tensors

Commutative Rings, by Irving Kaplansky. Revised edition. The University of Chicago Press, Chicago and London, 1974, ix+182pp.($9.75)

Symmetry Groups and their Applications, by Willard Miller, Jr. Academic Press.     

Representative of Quasidifferentials and Its Formula for a Quasidifferentiable Function

The quasidifferential of a quasidifferentiable function in the sense of Demyanov and Rubinov is not uniquely defined. Xia proposed the notion of the kernelled quasidifferential, which is expected to be a representative for the equivalent class of quasidifferentials. In the 2-dimensional case, the existence of the kernelled quasidifferential was shown. In this paper, the existence of the kernelled quasidifferential in the n-dimensional space (n>2) is proved under the assumption that the Minkowski difference and the Demyanov difference of subdifferential and minus superdifferential coincide. In particular, given a quasidifferential, the kernelled quasidifferential can be formulated. Applications to two classes of generalized separable quasidifferentiable functions are developed. Mathematics Subject Classifications (2000) 49J52, 54C60, 90C26. This work was supported by Shanghai Education Committee (04EA01).     

Zur lösung einiger minimax-aufgaben über dem quadrat

The paper deals with the minimax problems (2), (3) for probability distribution functions G or bounded measurable functions g and bounded measurable or quadratic summable functions f with the restrictions (6), (7) and (8), respectively. Under suitable assumptions apart from elementary cases the optimal solutions G or g and f can be calculated by means of certain Fredholm integral equations.     

THE EIGENVALUE PERTURBATION BOUND FOR ARBITRARY MATRICES

In this paper we present some new absolute and relative perturbation bounds for theeigenvalue for arbitrary matrices, which improves some recent results. The eigenvalueinclusion region is also discussed.     

Safety neighbournoods for the invariants of the matrix similarity

We give safety neighbourhoods for the necessary conditions in the change of the Jordan canonical form of a matrix under small perturbations. We also obtain the minimum distance from an n × n complex matrix which has less than k nonconstant invariant factors (2≤ k≤ n) to the set of matrices which have more or equal to k. When k= 2, we get in particular the distance from a nonderogatory matrix to the set of derogatory matrices.     

On the eigenvalues of specially low-rank perturbed matrices

We study the eigenvalues of a matrix A perturbed by a few special low-rank matrices. The perturbation is constructed from certain basis vectors of an invariant subspace of A, such as eigenvectors, Jordan vectors, or Schur vectors. We show that most of the eigenvalues of the low-rank perturbed matrix stayed unchanged from the eigenvalues of A; the perturbation can only change the eigenvalues of A that are related to the invariant subspace. Existing results mostly studied using eigenvectors with full column rank for perturbations, we generalize the results to more general settings. Applications of our results to a few interesting problems including the Google’s second eigenvalue problem are presented.     

Spectral properties of sums of certain Kronecker products

We introduce two kinds of sums of Kronecker products, and their induced operators. We study the algebraic properties of these two kinds of matrices and their associated operators; the properties include their eigenvalues, their eigenvectors, and the relationships between their spectral radii or spectral abscissae. Furthermore, two projected matrices of these Kronecker products and their induced operators are also studied.     

有理标准形存在性定理的构造性证明

对抽象代数中的一个基本定理有理标准形存在性定理给出了一个构造性证明,用直观的向量形式解释了Frobenius基的结构和生成Frobenius基的向量所需的条件.     

A matrix generation approach for eigenvalue optimization

We study the extension of a column generation technique to nonpolyhedral models. In particular, we study the problem of minimizing the maximum eigenvalue of an affine combination of symmetric matrices. At each step of the algorithm a restricted master problem in the primal space, corresponding to the relaxed dual (original) problem, is formed. A query point is obtained as an approximate analytic center of a bounded set that contains the optimal solution of the dual problem. The original objective function is evaluated at the query point, and depending on its differentiability a column or a matrix is added to the restricted master problem. We discuss the issues of recovering feasibility after the restricted master problem is updated by a column or a matrix. The computational experience of implementing the algorithm on randomly generated problems are reported and the cpu time of the matrix generation algorithm is compared with that of the primal-dual interior point methods on dense and sparse problems using the software SDPT3. Our numerical results illustrate that the matrix generation algorithm outperforms primal-dual interior point methods on dense problems with no structure and also on a class of sparse problems. This work has been completed with the partial support of a summer grant from the College of Business Administration, California State University San Marcos, and the University Professional Development/Research and Creative Activity Grant     

A note on strong pseudoconvexity

A strongly pseudoconvex function is generalized to non-smooth settings. A complete characterization of the strongly pseudoconvex radially lower semicontinuous functions is obtained.      

Existence of minimizers for spectral problems

In this paper we show that any increasing functional of the first k   eigenvalues of the Dirichlet Laplacian admits a (quasi-)open minimizer among the subsets of R^N${\mathbb{R}}^N$ of unit measure. In particular, there exists such a minimizer which is bounded, where the bound depends on k and N, but not on the functional.     

Minimization of marginal functions in mathematical programming based on continuous outer subdifferentials

Typically, exact information of the whole subdifferential is not available for intrinsically nonsmooth objective functions such as for marginal functions. Therefore, the semismoothness of the objective function cannot be proved or is even violated. In particular, in these cases standard nonsmooth methods cannot be used. In this paper, we propose a new approach to develop a converging descent method for this class of nonsmooth functions. This approach is based on continuous outer subdifferentials introduced by us. Further, we introduce on this basis a conceptual optimization algorithm and prove its global convergence. This leads to a constructive approach enabling us to create a converging descent method. Within the algorithmic framework, neither semismoothness nor calculation of exact subgradients are required. This is in contrast to other approaches which are usually based on the assumption of semismoothness of the objective function.     

具有时滞的一般型脉冲神经网络的指数稳定性

讨论具有时滞的一般性脉冲神经网络的稳定性.在不假定激励函数有界或可导的前提下,利用非光滑分析和Lyapunov泛函,得到了这类神经网络系统平衡点的存在唯一性和全局指数稳定性判别准则.作为特例,得到了Hopfield神经网络,时滞细胞神经网络,双向联想记忆神经网络的平衡点的存在唯一性和全局指数稳定性判定定理.