群逆的扰动界及其在奇异线性系统中的应用

本文建立了群逆的扰动界,此界基于矩阵A的Jordan标准形和P-范数,其中P是非异矩阵满足P-1AP=[D000],D是非异上双对角阵且‖A‖P=‖P-1AP‖2.当矩阵A和A+E有相同的秩且‖E‖P较小时,得到了‖(A+E)#-A#‖P较好的估计.在相同的条件下,研究了相容的奇异线性系统Ax=b的扰动,给出了xopt=A#b扰动的上界,其中A#是A的群逆,xopt是最小P-范数解.     

酉不变范数下极分解的扰动界

设A是m×n(m≥n)且秩为n的复矩阵.存在m×n矩阵Q满足Q*Q=I和n×n正定矩阵H使得A=QH,此分解称为A的极分解.本文给出了在任意酉不变范数下正定极因子H的扰动界,改进文[1,11]的结果;另外也首次提供了乘法扰动下酉极因子Q在任意酉不变范数下的扰动界.     

关于矩阵群逆的逆序律

得到了体上两个n阶方阵A,B的群逆A#,B#若存在,则其乘积的群逆(AB) #也存在,且(AB) #=B#A#成立的充分与必要条件是:存在n阶可逆矩阵P使得A =Pdiag(A1,A2 ,…,As) P- 1,B =Pdiag(B1,B2 ,…,Bs) P- 1且对于任意i(i=1 ,2 ,…,s)有Ai,Bi阶数相同,Ai,Bi为可逆矩阵或为0矩阵;又对i≠1有Ai Bi=0 .     

次酉极因子在酉不变范数下的相对扰动界

设A是一个m×n阶复矩阵,分解A=QH称为广义极分解,如果Q是m×n次酉极因子且H为n×n半正定的Hermite矩阵．本文获得了次酉极因子在任意酉不变范数下的几个相对扰动界,在某种意义上,相对扰动界比R．C．Li等获得的绝对扰动界要好．     

一般矩阵函数的变差

设Sn是n次对称群,G为Sn的子群,χ是G的次数为1的特征标.如果A是一个n阶复变矩阵,定义一般矩阵函数dχG为dχG(A)=∑σ∈Gχ(σ)∏ni=1aiσ(i).本文用lp-算子范数(1≤p≤∞)的性质证明了一般矩阵函数变差的两个不等式.     

利用静力试验数据修正具有广义中心对称有限元模型

设X,B分别是测得的位移矩阵和载荷矩阵,C是有限元方法得到的理论模型的估计,找广义中心对称的矩阵（A）使得（A）X=B,且是Frobenius范数意义下C的最佳逼近.并给出了解（A）的扰动分析,数值结果表明该方法是行之有效的.     

EP矩阵与矩阵方程的解

证明了如下结论:设A∈C~(n×n)是群可逆矩阵,则(i)A为EP矩阵当且仅当矩阵方程A~HXA=XAA~H在χ_A至少有一个解;(ii)A为EP矩阵当且仅当矩阵方程A~HXA=AA~HX在χ_A至少有一个解,其中χ_A={A,A~#,A~+,A~H,(A~#)~H,(A~+)~H}.     

矩阵SR分解R因子新的扰动界

矩阵的SR分解是求解一些优化控制问题的有效工具,如用来求解代数Riccati方程.利用分块的矩阵-向量方程方法与Lyapunov控制函数和Banach不动点定理相结合的方法获得了SR分解R因子在范数型扰动下的范数型的严格扰动界和一阶扰动界,改进了已有结果.     

Hermite矩阵与可对称化矩阵特征值之间的扰动上界

正1引言矩阵特征值的扰动问题,就是研究矩阵元素的改变对矩阵特征值的影响.设矩阵A,B为n阶复矩阵,矩阵B为矩阵A经过扰动之后的矩阵,且λ(A)={λ_i},λ(B))={μ_i},研究矩阵特征值的扰动就是研究λ(A)与λ(B)之间的差距,一般用2范数和Frobenius范数来描述它们之间的差距.矩阵特征值问题是由于处理数据时存在误差而引起的,使得到的特征值往往是经过     

矩阵广义逆新的扰动上界

利用矩阵的奇异值分解方法,研究了矩阵广义逆的扰动上界,得到了在F-范数下矩阵广义逆的扰动上界定理,所得定理推广并彻底改进了近期的相关结果.相应的数值算例验证了定理的有效性.     

关于矩阵方程X+A*X-1A=P的解及其扰动分析

考虑非线性矩阵方程X＋A^＊（X^-1）A=P其中A是n阶非奇异复矩阵,P是n阶Hermite正定矩阵．本文给出了Hermite正定解和最大解的存在性以及获得最大解的一阶扰动界,改进了文[5,6]中的部分结论．     

A PERTURBATION ANALYSIS FOR THE PROJECTION OF A STIFFLY SCALED MATRIX

In this paper we study the perturbation bound of the projection (WA )(WA ), where both the matrices A and W are given with W positive diagonal and severely stiff. When the perturbed matrix A=A δA satisfy several row rank preserving conditions, we derive a new perturbation bound of the projection.     

ON THE MINIMAL NONNEGATIVE SOLUTION OF ONSYMMETRIC ALGEBRAIC RICCATI EQUATION

We study perturbation bound and structured condition number about the minimalnonnegative solution of nonsymmetric algebraic Riccati equation,obtaining a sharp per-turbation bound and an accurate condition number.By using the matrix sign functionmethod we present a new method for finding the minimal nonnegative solution of this al-gebraic Riccati equation.Based on this new method,we show how to compute the desiredM-matrix solution of the quadratic matrix equation X~2-EX-F=0 by connecting itwith the nonsymmetric algebraic Riccati equation,where E is a diagonal matrix and F isan M-matrix.     

On the Minimal Nonnegative Solution of Nonsymmetric Algebraic Riccati Equation

We study perturbation bound and structured condition number about the minimal nonnegative solution of nonsymmetric algebraic Riccati equation, obtaining a sharp perturbation bound and an accurate condition number. By using the matrix sign function method we present a new method for finding the minimal nonnegative solution of this algebraic Riccati equation. Based on this new method, we show how to compute the desired M-matrix solution of the quadratic matrix equation X^2 - EX - F = 0 by connecting it with the nonsymmetric algebraic Riccati equation, where E is a diagonal matrix and F is an M-matrix.     

矩阵方程X-A*X-2A=Q的正定解及其扰动分析

研究非线性矩阵方程X-A*X-2A=Q(Q>0)的Hermite正定解及其扰动问题.给出了该方程存在唯-Hermite正定解的充分条件及解的迭代计算公式.在此条件下,给出了该唯一解的扰动界及正定解条件数的一种表达式,并用数值例子对所得结果进行了说明.     

ON THE ACCURACY OF THE LEAST SQUARES AND THE TOTAL LEAST SQUARES METHODS

Consider solving an overdetermined system of linear algebraic equations by both the least squares method (LS) and the total least squares method (TLS). Extensive published computational evidence shows that when the original system is consistent. one often obtains more accurate solutions by using the TLS method rather than the LS method. These numerical observations contrast with existing analytic perturbation theories for the LS and TLS methods which show that the upper bounds for the LS solution are always smaller than the corresponding upper bounds for the TLS solutions. In this paper we derive a new upper bound for the TLS solution and indicate when the TLS method can be more accurate than the LS method.Many applied problems in signal processing lead to overdetermined systems of linear equations where the matrix and right hand side are determined by the experimental observations (usually in the form of a lime series). It often happens that as the number of columns of the matrix becomes larger, the ra     

矩阵方程X—A*X~qA=I(0＜q＜1)Hermite正定解的扰动分析

首先证明了非线性矩阵方程X-A~*X~qA=I(0相似文献   

矩阵方程X-A~*X~(-1)A=Q的Hermite正定解及其扰动分析

考虑非线性矩阵方程X-A*X-1A=Q,其中A是n阶复矩阵,Q是n阶Hermite正定解,A*是矩阵A的共轭转置.本文证明了此方程存在唯一的正定解,并推导出此正定解的扰动边界和条件数的显式表达式.以上结果用数值例子加以说明.     

THE STRUCTURE SENSITIVITY ANALYSIS FOR TRIANGULAR SYSTEMS

Triangular systems play a fundamental role in matrix computations. It has become commonplace that triangular systems are solved to be more accurate even if they are ill-conditioned. In this paper, we define structured condition number and give structured (forward) perturbation bound. In addition, we derive the representation of optimal structured backward perturbation bound.     

矩阵方程X+A~*X~(-n)A=P的Hermite正定解及其扰动分析

考虑非线性矩阵方程X A~*X~(-n)A=P,其中A是m阶非奇异复矩阵,P是m阶Hermite正定矩阵.本文利用不动点理论讨论了该方程Hermite正定解的存在性及包含区间,给出了极大解的性质及求极大,极小解的迭代算法.研究了极大解的扰动问题,利用微分等方法获得了两个新的一阶扰动界,并给出数值例子对所得结果进行了比较说明.