Oppenheim不等式的一种类似

证明了实对称正定矩阵或实对称半正定矩阵与 M-矩阵的 Hadamard乘积满足实对称正定矩阵 Hadamard乘积的 Oppenheim不等式 .     

关于正定厄米特矩阵的一个不等式的推广

本文推广了正定厄米特矩阵的一个不等式 ,得到以下结果 :设 A( i) ,B( i) ,… ,C( i) ( i=1 ,2 ,… ,m)都是 n阶正定厄米特矩阵 ,A( i)11,B( i)11,… ,C( i)11为其相应矩阵的 k阶顺序主子阵 ,1≤ k≤ n-1 ,α,β,… ,γ都是正实数 ,且 α+β+… +γ=p≥ 1 ,则有∑mi=1|A( i) |α|A( i)11|α,|B( i) |β|B( i)11|β… |C( i) |γ|C( i)11|γ) <∑mi=1A( i) α∑mi=1A( i)11α.∑mi=1B( i) β∑mi=1B( i)11β…∑mi=1C( i) γ∑mi=1C( i)11γ     

正项矩阵级数的敛散性研究

本文研究了正项矩阵级数收敛的充要条件 ,从而把正项级数的收敛原理推广到了正项矩阵级数的情形 .     

矩阵的酉不变范数不等式

利用凹函数和半正定矩阵的性质,讨论并且得到了一些矩阵Rotfel型范数不等式.另外,通过研究Hermitian矩阵和斜Hermitian矩阵和的特征值的模行列式的不等式,得到一些关于Hermitian矩阵和斜Hermitian矩阵和的范数不等式.推广了文献中的相关结果.     

广义正定矩阵的Oppenheim不等式

建立了PDn类广义正定矩阵的广义Oppenheim不等式,推广了以前的结果.     

复正定矩阵的充要条件

本文研究Hermite正定矩阵、实正定矩阵的推广概念复正定矩阵,给出了复方阵复正定性的一些充分必要条件。     

Necessary and Sufficient Condition for Generalized Diagonal Dominance Matrices

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完全主正矩阵的行列式不等式

给出了完全主正矩阵的凸性不等式和Minkowski型不等式,并推出了M矩阵,亚正定矩阵等类型的矩阵在一定条件下的凸性不等式和Minkowski型不等式.     

Modified Hermitian and skew‐Hermitian splitting methods for non‐Hermitian positive‐definite linear systems

To further study the Hermitian and non‐Hermitian splitting methods for a non‐Hermitian and positive‐definite matrix, we introduce a so‐called lopsided Hermitian and skew‐Hermitian splitting and then establish a class of lopsided Hermitian/skew‐Hermitian (LHSS) methods to solve the non‐Hermitian and positive‐definite systems of linear equations. These methods include a two‐step LHSS iteration and its inexact version, the inexact Hermitian/skew‐Hermitian (ILHSS) iteration, which employs some Krylov subspace methods as its inner process. We theoretically prove that the LHSS method converges to the unique solution of the linear system for a loose restriction on the parameter α. Moreover, the contraction factor of the LHSS iteration is derived. The presented numerical examples illustrate the effectiveness of both LHSS and ILHSS iterations. Copyright © 2007 John Wiley & Sons, Ltd.     

‘Modified Hermitian and skew‐Hermitian splitting methods for non‐Hermitian positive‐definite linear systems’ [Numer. Linear Algebra Appl. 14 (2007) 217–235]

In this note, some errors in the article (Numer. Linear Algebra Appl. 2007; 14 :217–235) are pointed out and some correct results are presented. Copyright © 2011 John Wiley & Sons, Ltd.     

关于一Bellman不等式的一点注记(英文)

考虑不等式 :tr(AB) m≤ tr(Am Bm) ,m=1 ,2 ,3 ,… ,其中矩阵 A,B均为 n× n(n为任意的自然数 )的实对称正定矩阵 .它是 Richard Bellman教授在 1 980年德国 Oberwolfach市召开的第二届国际不等式会议上提出的 2 0个矩阵迹不等式的其中之一 .其余 1 9个不等式均被彻底解决 .本文给出了一个有效的使得上述不等式成立的充分条件     

非奇异M矩阵的判定及并行算法

1　引　　言M矩阵是一类具有非正非对角元和非负对角元的实方阵 .M矩阵在生物学、经济学、智能科学、计算方法等许多学科中都有重要应用 .许多实际问题的应用都归到 M矩阵的判定上 .例如判定一个矩阵是否为 M矩阵在网络计算中可以判定一个离散动力系统是否稳定 .在数值计算中 ,可以判定一个迭代系统是否收敛 .因此研究 M矩阵的判定方法成为矩阵理论研究中极为活跃的一个领域 .目前国内外许多数学工作者都在研究 M矩阵的判定方法 ,已有的研究成果都是对 M矩阵的整体进行讨论 ,这对高阶矩阵来说 ,不仅计算困难 ,而且需要对判定定理进行消化…     

M-矩阵的性质与Hadamard-Fisher不等式的注记

主要研究了两个M-矩阵的比较性质与不等式,给出了M-矩阵与逆M-矩阵Hadamard-Fisher不等式等式成立的矩阵结构.     

Improved convergence theorems for new Hermitian and skew-Hermitian splitting methods

In this note, based on the previous work by Pour and Goughery (New Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems Numer. Algor. 69 (2015) 207–225), we further discuss this new Hermitian and skew-Hermitian splitting (described as NHSS) methods for non-Hermitian positive definite linear systems. Some new convergence conditions of the NHSS method are obtained, which are superior to the results in the above paper.     

On the Hermitian positive definite solutions of nonlinear matrix equation X + A∗XA = Q

Nonlinear matrix equation X^s + A∗X^−tA = Q, where A, Q are n × n complex matrices with Q Hermitian positive definite, has widely applied background. In this paper, we consider the Hermitian positive definite solutions of this matrix equation with two cases: s ? 1, 0 < t ? 1 and 0 < s ? 1, t ? 1. We derive necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions for the matrix equation and obtain some properties of the solutions. We also propose iterative methods for obtaining the extremal Hermitian positive definite solution of the matrix equation. Finally, we give some numerical examples to show the efficiency of the proposed iterative methods.     

Positive definite solutions of the nonlinear matrix equation X + AXA = Q (q > 0)

In this paper, the nonlinear matrix equation X + A^∗X^qA = Q (q > 0) is investigated. Some necessary and sufficient conditions for existence of Hermitian positive definite solutions of the nonlinear matrix equations are derived. An effective iterative method to obtain the positive definite solution is presented. Some numerical results are given to illustrate the effectiveness of the iterative methods.     

三对角的完全非负矩阵上的Schur-Oppenheim严格不等式

应用完全非负矩阵的 Hadamard中心的性质 ,给出了非奇异三对角完全非负矩阵的Hadamard乘积的行列式的下界估计满足 Schur- Oppenheim严格不等式的充分条件 ,改进了 T.L .Markham的关于三对角的振荡矩阵的相应结果 .     

一类亚正定矩阵上的逆向Hadamard不等式和逆向Szasz不等式

利用亚正定矩阵的基本理论,建立了一类亚正定矩阵上的逆向Hadamard不等式和逆向Szasz不等式.