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1.
线性流形上的广义中心对称矩阵反问题   总被引:4,自引:0,他引:4  
袁永新  戴华 《计算数学》2005,27(4):383-394
设R∈Cn×n是满足R=RH=R-1≠±In的广义反射矩阵.若A∈Cn×n满足RAR=A,则称A为n阶广义中心对称矩阵,n阶广义中心对称矩阵的全体记为GCSCn×n.令X1,Z1∈Cn×k1,Y1,W1∈Cn×l1,S={A|‖AX1-Z1‖2+‖Y1HA-W1H‖2=min,A∈GCSCn×n},本文研究如下问题.问题Ⅰ.给定矩阵Z2,X2∈Cn×k2,Y2,W2∈Cn×l2,求A∈S,使得其中‖·‖是Frobenius范数.问题Ⅱ.给定矩阵A∈Cn×n,求A∈SE,使得其中SE是问题Ⅰ的解集合.本文给出了问题Ⅰ解集合SE的表达式,并导出了矩阵方程AX2=Z2,Y2HA=W2H有解A∈S的充分必要条件及其通解表达式,并给出了问题Ⅱ解的表达式以及求解问题Ⅱ的数值方法和数值例子.  相似文献   

2.
讨论了线性流形上广义中心对称矩阵的最小二乘解,得到了解的一般表达式。对于任意给定的实对称矩阵A,在最小二乘解集中得到了A的最佳逼近解.  相似文献   

3.
利用反埃尔米特广义汉密尔顿矩阵的表示定理,得到了线性流形上反埃尔米特广义汉密尔顿矩阵反问题的最小二乘解的一般表达式,建立了线性矩阵方程在线性流形上可解的充分必要条件.对于任意给定的n阶复矩阵,证明了相关最佳逼近问题解的存在性与惟一性,并推得了最佳逼近解的表达式.  相似文献   

4.
线性约束下Hermite-广义反Hamilton矩阵的最佳逼近问题   总被引:3,自引:0,他引:3  
本文利用对称向量与反对称向量的特征性质,给出了约束矩阵集合非空的充分必要条件及矩阵的一般表达式.运用空间分解理论和闭凸集上的逼近理论,得到了任一n阶复矩阵在约束矩阵集合中的惟一最佳逼近解.  相似文献   

5.
线性流形上Hermite-广义反Hamilton矩阵反问题的最小二乘解   总被引:8,自引:0,他引:8  
张忠志  胡锡炎  张磊 《计算数学》2003,25(2):209-218
1.引言 令Rn×m表示所有n×m实矩阵集合,Cn×m表示所有n×m复矩阵集合,Cn=Cn×1,HCn×n表示所有n阶Hermite矩阵集合,UCn×n表示所有n阶酉矩阵集合,AHCn×n表示所有n阶反Hermite矩阵集合,R(A)表示A的列空间,N(A)表示A的零空间,A+表示A的Moore—Penrose广义逆,A*B表示A与B的Hadamard积,rank(A)表示矩阵A的秩.tr(A)表示矩阵A的迹.矩阵A,B的内积定义为(A,B)=tr(BHA),A,B∈Cn×m,由此内积诱导的范数为||A||=√(A,A)=[tr(AHA)]1/2,则此范数为Frobenius范数,并且Cn×m构成一个完备的内积空间,In表示n阶单位阵,i=√-1,记OASRn×n表示n×n阶正交反对称矩阵的全体,即  相似文献   

6.
研究线性流形上广义次对称矩阵的左右逆特征值问题及其最佳逼近问题.利用广义次对称矩阵的性质及矩阵的奇异值分解得到问题的通解表达式.同时,给出其有唯一的最佳逼近解以及求最佳逼近解的算法.  相似文献   

7.
线性流形上D对称矩阵反问题的最小二乘解   总被引:3,自引:0,他引:3  
本研究了线性流形上D对称矩阵反问的最小二乘解及其逼近问题,给出了最小二乘解的一般表达式,并就该问题的特殊情况-矩阵反问题,获得了有解的充分必要条件,并在有解的条件下得到了解的一段表达式。  相似文献   

8.
本讨论在线性流形上广义反对称矩阵的最佳逼近,给出了若干有意义的结果。  相似文献   

9.
线性流形上的矩阵最佳逼近   总被引:7,自引:1,他引:7  
令S={A∈Rn×m|f1(A)=‖AX1-Z1‖2+‖YT1A-WT1‖2=min},其中X1∈Rm×k1,Z1∈Rn×k1,Y1∈Rn×11和W1∈Rm×11均为给定的矩阵,‖·‖是Frobenius范数。本文考虑如下问题:问题Ⅰ给定X2∈Rm×k2,Z2∈Rn×k2,Y2∈Rn×l2,W2∈Rm×l2,求A∈S,使得f2(A)=‖AX2-Z2‖2+‖YT2A-WT2‖2=min.问题Ⅱ给定A∈Rn×m,求A∈SA,使得‖A-A‖=infA∈SA‖A-A‖,其中SA是问题I的解集合。本文给出问题I解集合SA的通式和问题Ⅱ的解A的表达式,提出了求解问题Ⅰ与Ⅱ的数值方法。许多文献的结果都是本文结果的特例。  相似文献   

10.
反中心对称矩阵的广义特征值反问题   总被引:8,自引:0,他引:8  
Given matrix X and diagonal matrix A , the anti-centrosymmetric solutions (A, B) and its optimal approximation of inverse generalized eigenvalue problem AX = BXA have been considered. The general form of such solutions is given and the expression of the optimal approximation solution to a given matrix is derived. The algorithm and one numerical example for solving optimal approximation solution are included.  相似文献   

11.
该文讨论了线性流形上矩阵方程AX=B反对称正交对称反问题的最小二乘解及其最佳逼近问题.给出了最小二乘问题解集合的表达式,得到了给定矩阵的最佳逼近问题的解,最后给出计算任意矩阵的最佳逼近解的数值方法及算例.  相似文献   

12.
Let S∈Rn×n be a symmetric and nontrival involution matrix. We say that A∈E R n×n is a symmetric reflexive matrix if AT = A and SAS = A. Let S R r n×n(S)={A|A= AT,A = SAS, A∈Rn×n}. This paper discusses the following two problems. The first one is as follows. Given Z∈Rn×m (m < n),∧= diag(λ1,...,λm)∈Rm×m, andα,β∈R withα<β. Find a subset (?)(Z,∧,α,β) of SRrn×n(S) such that AZ = Z∧holds for any A∈(?)(Z,∧,α,β) and the remaining eigenvaluesλm 1 ,...,λn of A are located in the interval [α,β], Moreover, for a given B∈Rn×n, the second problem is to find AB∈(?)(Z,∧,α,β) such that where ||.|| is the Frobenius norm. Using the properties of symmetric reflexive matrices, the two problems are essentially decomposed into the same kind of subproblems for two real symmetric matrices with smaller dimensions, and then the expressions of the general solution for the two problems are derived.  相似文献   

13.
By using Moore-Penrose generalized inverse and the general singular value decomposition of matrices, this paper establishes the necessary and sufficient conditions for the existence of and the expressions for the centrosymmetric solutions with a submatrix constraint of matrix inverse problem AX = B. In addition, in the solution set of corresponding problem, the expression of the optimal approximation solution to a given matrix is derived.  相似文献   

14.
Traditionally an inverse eigenvalue problem is about reconstructing a matrix from a given spectral data. In this work we study the set of real matrices A of order n such that the linear complementarity system
  相似文献   

15.
The implicit Q theorem for Hessenberg matrices is a widespread and powerful theorem. It is used in the development of, for example, implicit QR algorithms to compute the eigendecomposition of Hessenberg matrices. Moreover it can also be used to prove the essential uniqueness of orthogonal similarity transformations of matrices to Hessenberg form. The theorem is also valid for symmetric tridiagonal matrices, proving thereby also in the symmetric case its power. Currently there is a growing interest to so-called semiseparable matrices. These matrices can be considered as the inverses of tridiagonal matrices. In a similar way, one can consider Hessenberg-like matrices as the inverses of Hessenberg matrices. In this paper, we formulate and prove an implicit Q theorem for the class of Hessenberg-like matrices. We introduce the notion of strongly unreduced Hessenberg-like matrices and also a method for transforming matrices via orthogonal transformations to this form is proposed. Moreover, as the theorem is valid for Hessenberg-like matrices it is also valid for symmetric semiseparable matrices. The research was partially supported by the Research Council K.U.Leuven, project OT/00/16 (SLAP: Structured Linear Algebra Package), by the Fund for Scientific Research–Flanders (Belgium), projects G.0078.01 (SMA: Structured Matrices and their Applications), G.0176.02 (ANCILA: Asymptotic aNalysis of the Convergence behavior of Iterative methods in numerical Linear Algebra), G.0184.02 (CORFU: Constructive study of Orthogonal Functions) and G.0455.0 (RHPH: Riemann-Hilbert problems, random matrices and Padé-Hermite approximation), and by the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister’s Office for Science, Technology and Culture, project IUAP V-22 (Dynamical Systems and Control: Computation, Identification & Modelling). This research was partially supported by by MIUR, grant number 2004015437 (third author). The scientific responsibility rests with the authors.  相似文献   

16.
17.
有许多类直接控制系统的绝对稳定性[1]涉及到这样一类线性方程组协Ax=b的反问题:对于给定的x,b∈Rn,n阶实矩阵类Ⅱ(n),求解集Ⅰ(Ⅱ(n);x,b)={A∈Ⅱ(n)|Ax=b}非空的条件.文[2]讨论了反问题Ⅰ(Ps(n);x,b)≠(Ps(n)为正定降类)和Ⅰ(O(n);x,b)≠(O(n)为正交阵类)的条件,文[3]进一步给出了Ⅰ(M-阵类;x,b)和Ⅰ(S-阵类;x,b)有解的条件.本文将研究这类反问题在更广的一类矩阵类─—广义正定矩阵[4,5]类中的求解,从而使这类反问题得到了较完满的解决.  相似文献   

18.
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