差分代换矩阵与多项式的非负性判定

主要分析了差分代换矩阵的基本性质,证明了存在有限个差分代换矩阵的乘积可以将单位点$(1,0,\cdots,0)$变换到指定的非负(本原)整点.利用这一结果可以导出${R^n_+}$上判定半正定型的充要条件.根据此充要条件建立的算法(TSDS)可能不停机,针对不停机的情况,再给出一些判定半正定型的充分条件.     

关于圈的完全正矩阵实现

本文给出了一个关联图为圈的非负、半正定矩阵A为完全正的一个充要条件.我们还证明了这样的矩阵A（当A为完全正时）的分解指数即为A的阶数.     

关于HADAMARD不等式的注记

本文主要研究一类Ｆ－矩阵的性质，这类矩阵包含对称半正定矩阵，完全非负矩阵，τ矩阵和Ｍ－矩阵为其子类。我们不仅对Ｆ－矩阵改进了Ｈｄａｍａｒｄ不等式，而且证明对此类矩阵Ｈａｄａｍａｒｄ不等式成立等式的充要条件是它的每条对角线，除主对角线外，都含有零元。     

实正定矩阵的复合矩阵的正定性

本文讨论了实正定矩阵的复合矩阵的正定性，并且给出了实正定矩阵的复合矩阵仍为正定矩阵的一个充要条件.     

矩阵方程A^TXB=C的正定和半正定解

给出了矩阵方程Ａ＾ＴＸＢ＝Ｃ在正定和半正定矩阵类中有解的充要条件及解的一般表达式。     

多元线性混合模型方差分量矩阵的非负估计

本文研究了带有两个方差分量矩阵的多元线性混合模型方差分量矩阵的估计问题.对于平衡模型,给出了基于谱分解估计的一个方差分量矩阵的非负估计类.对于非平衡模型,给出了方差分量矩阵的广义谱分解估计类,讨论了与ANOVA估计等价的充要条件.同时,在广义谱分解估计的基础上给出了一种非负估计类,并讨论了其优良性.当具有较小二次风险的非负估计不存在时,从估计为非负的概率的角度考虑,将Kelly和Mathew(1993)提出的构造具有更小取负值概率的估计类的方法推广到本文的多元模型下,给出了较谱分解估计相比有更小取负值概率和更小风险的估计类.最后,模拟研究和实例分析表明文中理论结果有很好的表现.     

关于正定矩阵的几个问题

本文推广了文［１］的主要定理，给出了用低阶矩阵判定高阶矩阵正定的判定定理，同时给出了矩阵方程ＡＸ＝Ｂ的反问题在正定矩阵类中解存在的充要条件及解的一般形式．     

正定矩阵的子式阵仍是正定矩阵的一个充要条件

文[2]证明了实对称正定矩阵的子式阵仍然是实对称正定矩阵,文[3]给出了一般的正定矩阵的的概念,本文利用标准型给出了一般正定矩阵的子式阵仍然是正定矩阵的充要条件.     

关于实方阵的正定性与规范性的进一步拓广

对亚正定矩阵类与实规范矩阵类作进一步拓广 ,得到一个新的矩阵类——亚规范矩阵 .研究了它的等价条件 ,合同下的标准形及有关行列式的性质 ,所得结果推广并改进了亚正定矩阵与实规范矩阵的一些结论 .     

关于线性代数方程组 AX=b 反问题的再讨论

本文研究线性代数方程组 AX=b 的反问题,在实对称半正定(半负定)矩阵类中有解的充要条件和几个重要推论.     

非负定阵元素的扰动和正定完备化

在保持非负定性不变的前提下,本文对矩阵每一元素容许多大的扰动作了进一步的研究, 将本文的结论和C．R．Johnson提出的部分正定阵的正定完备化进行比较,容易发现对已知的正定矩阵求扰动,本文的结论比用C．R.Johnson的正定完备化计算扰动形式上更简单,同时也给出了不同于C．R．Johnson的部分正定阵的正定完备化表示的另外一个公式,推出了这些正定完备化矩阵应具有的若干性质．     

泛正定阵与广义正定阵的关系

本文利用矩阵的稳定值,给出泛正定阵的等价条件,进一步得到泛正定阵与广义正定阵之间的一些关系．     

Complete controllability and contractibility in multimodal systems

The concept of a strictly positive definite set of Hermitian matrices is introduced. It is shown that a strictly positive definite set is always a positive definite set, and conditions are found under which a positive definite set is strictly positive definite. We also show that a set of Hermitian matrices is strictly positive definite if and only if some nonnegative linear combination of these matrices is a positive definite matrix. For state dimension two, we use this concept to find necessary and sufficient conditions for a two-mode completely controllable irreducible multimodal system to be contractible relative to an elliptic norm. For general state dimensions, we give necessary and sufficient conditions for a special-type two-mode completely controllable irreducible system to be contractible relative to a weakly monotone norm. Applying the above results, we show that, for state dimension two, there exists a completely controllable two-mode system which is not contractible relative to either an elliptic or a weakly monotone norm. We leave open the question whether or not complete controllability implies contractibility, relative to some norm, for multimodal systems of two or more modes.     

Convexity of the inverse and Moore-Penrose inverse

Convexity properties of the inverse of positive definite matrices and the Moore-Penrose inverse of nonnegative definite matrices with respect to the partial ordering induced by nonnegative definiteness are studied. For the positive definite case null-space characterizations are derived, and lead naturally to a concept of strong convexity of a matrix function, extending the conventional concept of strict convexity. The positive definite results are shown to allow for a unified analysis of problems in reproducing kernel Hilbert space theory and inequalities involving matrix means. The main results comprise a detailed study of the convexity properties of the Moore-Penrose inverse, providing extensions and generalizations of all the earlier work in this area.     

关于复矩阵乘积的正定性

两复正定矩阵之和必是复正定矩阵,但其积未必是复正定矩阵.研究了复矩阵之积的正定性,给出了复矩阵之积为复正定矩阵的一系列判定条件,获得了一些新的结果,改进并推广了K y Fan T aussky定理及Fe jer定理.     

关于矩阵的Sharp序、*序和减序

给出短阵sharp序的一个新的刻画，由此得到(半)正定短阵sharp序与其平方矩阵sharp序之间的关系．我们还讨论正规矩阵的*序与减序之间的关系，推广了关于Hermmite矩阵的相应结果．     

State models and asymptotic behaviour of 2D positive systems

Homogeneous 2D positive systems are 2D state-space models whosevariables are alwalys nonnegative and, consequently, are describedby a pair of nonnegative square matrices (A, B). In the paper,the properties of these pairs are discussed both in the generalcase and under particular assumptions like finite memory, separability,and property L. Various aspects of the positive asymptotic dynamic are considered;in particular, sufficient conditions are provided guaranteeingthat the local states are eventually strictly positive. Finally,some results on the convergence of the states towards a constantasymptotic distribution are presented.     

Shorted matrices—an extended concept and some applications

The shorting of an operator, hitherto considered by Krein [11] and by Anderson and Trapp [3] only for positive operators, is extended to rectangular matrices and square matrices not necessarily hermitian nonnegative definite. Some applications of the shorted matrix in mathematical statistics are discussed.     

Computing a Nearest P-Symmetric Nonnegative Definite Matrix Under Linear Restriction

Let $P$ be an $n\times n$ symmetric orthogonal matrix. A real $n\times n$ matrix $A$ is called P-symmetric nonnegative definite if $A$ is symmetric nonnegative definite and $(PA)^T=PA$. This paper is concerned with a kind of inverse problem for P-symmetric nonnegative definite matrices: Given a real $n\times n$ matrix $\widetilde{A}$, real $n\times m$ matrices $X$ and $B$, find an $n\times n$ P-symmetric nonnegative definite matrix $A$ minimizing $||A-\widetilde{A}||_F$ subject to $AX =B$. Necessary and sufficient conditions are presented for the solvability of the problem. The expression of the solution to the problem is given. These results are applied to solve an inverse eigenvalue problem for P-symmetric nonnegative definite matrices.     

Parallel multisplitting two-stage iterative methods with general weighting matrices for non-symmetric positive definite systems

In this work, we propose a new parallel multisplitting iterative method for non-symmetric positive definite linear systems. Based on optimization theory, the new method has two great improvements; one is that only one splitting needs to be convergent, and the other is that the weighting matrices are not scalar and nonnegative matrices. The convergence of the new parallel multisplitting iterative method is discussed. Finally, the numerical results show that the new method is effective.