Heinz-type numerical radii inequalities

The main aim of this article is to present inequalities for the numerical radius of commutators of positive matrices. Some of these inequalities are analogues of known inequalities for unitarily invariant norms. In particular, variants, but weaker forms, of the well-known Heinz inequality and its generalizations are extended to the context of numerical radius.     

A convex treatment of numerical radius inequalities

We prove an inner product inequality for Hilbert space operators. This inequality will be utilized to present a general numerical radius inequality using convex functions. Applications of the new results include obtaining new forms that generalize and extend some well known results in the literature, with an application to the newly defined generalized numerical radius. We emphasize that the approach followed in this article is different from the approaches used in the literature to obtain such versions.

     

矩阵张量积与诱导矩阵的y-数值半径

从y-数值半径的定义出发,利用矩阵张量积与诱导矩阵的性质,研究了它们的y-数值半径,得到了矩阵张量积与诱导矩阵y-数值半径的几个不等式.     

New stability criteria for a class of neutral systems with discrete and distributed time-delays: an LMI approach

In this paper, a problem of the asymptotic stability for a class of neutral systems with multiple discrete and distributed time-delays is considered. Lyapunov stability theory is applied to guarantee the stability for the systems. New discrete-delay-independent and discrete-delay-dependent stability conditions are derived in terms of the spectral radius and linear matrix inequality. By mathematical analysis, the stability criteria are proved to be less conservative than the ones reported in the current literatures. A numerical example is given to illustrate the availability of the proposed results.     

A Buzano type inequality for two Hermitian forms and applications

A Buzano type inequality for two nonnegative Hermitian forms is obtained. Applications to inequalities for norm and numerical radius of bounded linear operators in complex Hilbert spaces are given.     

On estimate for numerical radius of some contractions

For the numerical radius of an arbitrary nilpotent operator T on a Hilbert space H, Haagerup and de la Harpe proved the inequality , where n ≥ 2 is the nilpotency order of the operator T. In the present paper, we prove a Haagerup-de la Harpe-type inequality for the numerical radius of contractions from more general classes. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1335–1339, October, 2006.     

非正则图的最大特征值

通过对图的最大特征分量与顶点度之间的关系的刻画,得到了图的谱半径与参数最大度和次大度之间的不等关系,进而获得了简单连通非正则图的谱半径的若干上界.     

广义Nekrasov矩阵的充分条件

对任意给定的矩阵,从矩阵元素出发,利用定义构造特殊的正对角矩阵,结合不等式的放缩和数学归纳法,给出广义Nekrasov矩阵判定的三个充分条件,并用数值实例说明了所得结果的有效性.     

矩阵不等式约束下矩阵方程AX=B的双对称解

本文讨论矩阵不等式CXD≥E 约束下矩阵方程AX=B的双对称解,即给定矩阵A,B,C,D和 E, 求双对称矩阵X, 使得AX=B 和 CXD≥E, 其中CXD≥E表示矩阵CXD-E非负.本文将问题转化为矩阵不等式最小非负偏差问题,利用极分解理论给出了求其解的迭代方法,并结合相关矩阵理论说明算法的收敛性.最后给出数值算例验证算法的有效性.     

Reverses of Schwarz inequality in inner product spaces with applications

New reverses of the Schwarz inequality in complex inner products spaces with applications for bounded linear operators are given. Some Grüss' type inequalities and their applications for numerical radius and the operator norm are provided as well.     

A further extension of Rotfel’d theorem

We prove an inequality for concave functions and partitioned matrices whose numerical ranges lie in a sector. This complements a theorem by E.Y. Lee concerning the positive semi-definite case.     

广义严格对角占优矩阵的一组判定条件

以M-矩阵以及α-对角占优矩阵为工具,对0≤α≤1,借助Hlder不等式给出了广义严格对角占优矩阵以及非奇异M-矩阵的几则新的充分条件,拓广了近期的一些相关结果,并用数值例子说明这些结果的有效性.     

Y-数值半径的乘法因子

设A是一个n阶复数矩阵,y=(ξ1,…,ξn)是n维复数组,称ry(A)=max{|∑ξixi*Axi|∶xi*xi=1,xi∈Cn}为矩阵A的Y-数值半径,其中Cn表示复数域C上的n维向量空间.当y=(1,0,…,0)时,Y-数值半径变为标准数值半径r(A)=max{|x*Ax|∶x*x=1}.证得当sum from i=1 to n(ξi)≠0且ξi不都相等时,ry是广义矩阵范数,同时还讨论了ry的乘法因子.     

An extension of the Fiedler-Markham determinant inequality

We extend a determinant inequality of Fiedler and Markham [On a theorem of Everitt, Thompson, and de Pillis, Math Slovaca, 44 (1994), 441–444] to the class of matrices whose numerical range is contained in a sector. Moreover, some related results are also included.     

Alternating Direction Method for a Class of Sylvester Matrix Equations with Linear Matrix Inequality Constraint

This paper introduces an alternating direction method of multipliers (ADMM) for finding solutions to a class of Sylvester matrix equation AXB = E subject to a linear matrix inequality constraint CXD≥G. Preliminary convergence properties of ADMM are presented. Numerical experiments are performed to illustrate the feasibility and effectiveness of ADMM. In addition, some numerical comparisons with a recent algorithm are also given.     

非负矩阵谱半径的一个新界值

在Wielandt定理的基础上进行了推广,得到了一种估计非负矩阵谱半径的新方法,数值例子显示了新方法所得到的结果更为精确.     

时滞广义时变系统的容许性分析与镇定

研究了时滞广义时变系统的容许性与镇定性问题.首先,基于广义Lyapunov不等式、线性矩阵不等式和受限等价方法,建立时滞广义时变系统的Lyapunov不等式,将时滞广义时变系统的容许性问题转化为求解时滞广义时变系统的Lyapunov不等式问题,得到了系统容许的充分条件.然后,根据充分条件进一步研究了时滞广义时变系统的镇定问题,给出了状态反馈镇定器的设计方法.最后,通过数值算例验证了所得结论的有效性.     

State feedback stabilization of uncertain linear time‐delay systems: A nonlinear matrix inequality approach

A nonlinear matrix inequality is derived as a stabilizability condition of linear uncertain time‐delay systems. This inequality is seen as a less conservative one as well as efficient for numerical computation than the existing results as seen when solving by cone‐complementary algorithm. Copyright © 2010 John Wiley & Sons, Ltd.     

关于正定矩阵的两个不等式

通过Hermite矩阵的谱分解及一个改进的Young不等式,得到了关于正定矩阵的两个不等式,所得结果是对一些经典的矩阵不等式的进一步推广.最后,作为应用,给出了著名的Holder不等式和Minkowsi不等式的一种反向形式.     

具有变时滞的离散双向联想记忆神经网络的全局指数稳定性

利用Lyapunov函数和线性矩阵不等式(LMI),给出了判定一类双向联想记忆(BAM)神经网络模型的指数稳定的充分性条件.该条件去掉了以往论文中所要求的激活函数单调,可微分的条件,而且所得结果利用里的工具易于检测.并举例说明本文结果的有效性.