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).     

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.     

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.     

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 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.     

一个Jordan块的平方根矩阵

设J=Jm（λ）是一个特征值为λ的m阶Jordan块矩阵,则J能开平方的充要条件是m=1,或者m≥2时,λ≠0．且当m≥2,λ≠0,J的平方根矩阵恰有两个：±A,这里     

方阵的开方问题

运用Jordan标准形理论,完整解决了复数方阵、实数方阵能否开平方、开立方的问题.     

Spectral asymptotics for nonsmooth perturbations of the harmonic oscillator

We obtain the spectral asymptotics of a nonsmooth perturbation of the one-dimensional harmonic oscillator. We use the technique of perturbation theory which is based on an asymptotic presentation of the part of the kernel of the nonperturbed operator resolvent in some neighborhood of an eigenvalue.     

First and Second-Order Optimality Conditions for Convex Composite Multiobjective Optimization

Multiobjective optimization is a useful mathematical model in order to investigate real-world problems with conflicting objectives, arising from economics, engineering, and human decision making. In this paper, a convex composite multiobjective optimization problem, subject to a closed convex constraint set, is studied. New first-order optimality conditions for a weakly efficient solution of the convex composite multiobjective optimization problem are established via scalarization. These conditions are then extended to derive second-order optimality conditions.     

Jordan基的结构

用构造性方法证明了以同一个特征向量为终端向量的Jordan链从始端向量到终端向量可以有不同的最大长度,并给出了Jordan基中Jordan链的某些性质,进一步完善了Jordan标准形过渡矩阵求法的补充条件.     

A selfadjoint second order hyperbolic system

We consider a second order hyperbolic system of the type

(1)     

Jordan基的结构

用构造性方法证明了以同一个特征向量为终端向量的Jordan链从始端向量到终端向量可以有不同的最大长度,并给出了Jordan基中Jordan链的某些性质,进一步完善了Jordan标准形过渡矩阵求法的补充条件.     

Hamilton－Cayley定理的另一种证法

本文提出了线性代数中重要的Ｈａｍｉｌｔｏｎ－Ｃａｙｌｅｙ定理的另一种证明．