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1.
对称正交对称矩阵反问题的最小二乘解   总被引:18,自引:0,他引:18  
戴华 《计算数学》2003,25(1):59-66
Let P ∈ Rn×n be a symmetric orthogonal matrix. A∈Rn×n is called a symmetric orthogonal symmetric matrix if AT = A and (PA) T = PA. The set of all n × n symmetric orthogonal symmetric matrices is denoted by SRnxnp. This paper discusses the following problems: Problem I. Given X,B∈ Rn×m, find A ∈SRn×np such that||AX - B|| = min Problem II. Given A∈ Rn×n, find A∈SL such thatwhere ||·|| is the Frobenius norm, and SL is the solution set of Problem I.The general form of SL is given. The solvability conditions for the inverseproblem AX = B in SRn×nP are obtained. The expression of the solution toProblem II is presented.  相似文献   

2.
This paper is mainly concerned with solving the following two problems: Problem Ⅰ. Given X ∈ Rn×m, B . Rm×m. Find A ∈ Pn such thatwhereProblem Ⅱ. Given A ∈Rn×n. Find A ∈ SE such thatwhere F is Frobenius norm, and SE denotes the solution set of Problem I.The general solution of Problem I has been given. It is proved that there exists a unique solution for Problem II. The expression of this solution for corresponding Problem II for some special case will be derived.  相似文献   

3.
THE INVERSE PROBLEM FOR PART SYMMETRIC MATRICES ON A SUBSPACE   总被引:2,自引:0,他引:2  
In this paper, the following two problems are considered:Problem Ⅰ. Given S∈E Rn×p,X,B 6 Rn×m, find A ∈ SRs,n such that AX = B, where SR8,n = {A∈ Rn×n|xT(A - AT) = 0, for all x ∈ R(S)}.Problem Ⅱ. Given A* ∈ Rn×n, find A ∈ SE such that ||A-A*|| = minA∈sE||A-A*||, where SE is the solution set of Problem Ⅰ.The necessary and sufficient conditions for the solvability of and the general form of the solutions of problem Ⅰ are given. For problem Ⅱ, the expression for the solution, a numerical algorithm and a numerical example are provided.  相似文献   

4.
AbstractThis paper is mainly concerned with solving the following two problems: Problem I. Given X Cnxm, A = diag( 1, 2, ..... , m) Cmxm . Find A ABSRnxn such thatAX = XAwhere ABSRnxn is the set of all real n x n anti-bisymmetric matrices. Problem II. Given A RnXn. Find A SE such thatwhere || || is Frobenius norm, and SE denotes the solution set of Problem I.The necessary and sufficient conditions for the solvability of Problem I have been studied. The general form of SB has been given. For Problem II the expression of the solution has been provided.  相似文献   

5.
The main aim of this paper is to discuss the following two problems:λm)∈Hm×m, find A ∈ BSH≥n×n such that AX= X∧, where BSH≥n×n denotes the set of all n × n quaternion matrices which are bi-self-conjugate and nonnegative definite.Problem Ⅱ:Given B ∈ Hn×m, find -B∈SE such that ||B- B||Q = minA∈sE ||B - A||Q,necessary and sufficient conditions for SE being nonempty are obtained. The general form of elements in SE and the expression of the unique solution B of problem Ⅱ are given.  相似文献   

6.
Let A and C denote real n × n matrices. Given real n-vectors x1, ... ,xm, m ≤ n, and a set of numbers L = {λ1,λ2,... ,λm}. We describe (I) the set (?) of all real n × n bisymmetric positive seidefinite matrices A such that Axi is the "best" approximate to λixi, i = 1,2,...,m in Frobenius norm and (II) the Y in set (?) which minimize Frobenius norm of ||C - Y||.An existence theorem of the solutions for Problem I and Problem II is given and the general expression of solutions for Problem I is derived. Some sufficient conditions under which Problem I and Problem II have an explicit solution is provided. A numerical algorithm of the solution for Problem II has been presented.  相似文献   

7.
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   

8.
In this paper, we first consider the least-squares solution of the matrix inverse problem as follows: Find a hermitian anti-reflexive matrix corresponding to a given generalized reflection matrix J such that for given matrices X, B we have minA ||AX - B||. The existence theorems are obtained, and a general representation of such a matrix is presented. We denote the set of such matrices by SE. Then the matrix nearness problem for the matrix inverse problem is discussed. That is: Given an arbitrary A^*, find a matrix A E SE which is nearest to A^* in Frobenius norm. We show that the nearest matrix is unique and provide an expression for this nearest matrix.  相似文献   

9.
刘颖  马红平  苗正科 《东北数学》2008,24(4):311-318
For a symmetric sign pattern S1 the inertia set of S is defined to be the set of all ordered triples si(S) = {i(A) : A = A^T ∈ Q(S)} Consider the n × n sign pattern Sn, where Sn is the pattern with zero entry (i,j) for 1 ≤ i = j ≤ n or|i -j|=n- 1 and positive entry otherwise. In this paper, it is proved that si(Sn) = {(n1, n2, n - n1 - n2)|n1≥ 1 and n2 ≥ 2} for n ≥ 4.  相似文献   

10.
线性流形上实对称半正定阵的一类反问题   总被引:3,自引:0,他引:3  
1 引  言文中记Rn×m为所有n×m阶实阵集合,SRn×n为所有n阶实对称阵集合,Pn表示所有n阶实对称半正定阵集合,A≥0表示方阵A对称半正定.A+、R(A)、N(A)分别表示矩阵A的Moore-Penrose广义逆,列空间和零空间,‖·‖表示Froblnius范数.对于Z.Y∈Rn×k,令S={A∈Pn|AZ=Y,ZTY∈PK,R(YT)=R(YTZ)}(1.1)  现考虑如下问题:问题 给定X.B∈Rn×m,找A∈S,使得AX=B(1.2)  问题 给定A∈Rn×n,找A∈SE,使得‖A-A‖=infA∈SE‖A-A‖(1.3)其中SE是问题的解集合.问题与具有重要的应用背景,当Y=ZΛ,Λ=diag(λ1,λ2,…  相似文献   

11.
线性流形上对称正交对称矩阵逆特征值问题   总被引:2,自引:0,他引:2  
周富照  胡锡炎  张磊 《计算数学》2003,25(3):281-292
1.引言 令R~(n×m)表示所有n×m阶实矩阵集合;OR~(n×n)表示所有n阶正交矩阵全体;A~+表示A的Moore-penrose广义逆;I_к表示К阶单位阵;SR~(n×n)表示n阶实对称矩阵的全体;rank(A)表示A的秩;||·||是矩阵的Frobenius范数;对A=(a_(ij)),B=(b_(ij))∈R~(n×m),A*B表示A与B的Hadamard乘积,其定义为A*B=(a_(ij),b_(ij))。  相似文献   

12.
实对称五对角矩阵逆特征值问题   总被引:11,自引:1,他引:10  
1 引 言 对于n阶实对称矩阵A=(aij),r是一个正整数,且1≤r≤n-1,当|i-j|>r时,aij=0(i,j=1,2,…,n),至少有一个i使得ai,i+r≠0,则称矩阵A是带宽为2r+1的实对称带状矩阵.特别地,当r=1时,称A为实对称三对角矩阵;当r=2时,称A为实对称五对角矩阵. 实对称带状矩阵逆特征值问题应用十分广泛,这类问题不仅来自微分方程逆特征值问  相似文献   

13.
实对称矩阵广义特征值反问题   总被引:10,自引:0,他引:10  
本文研究如下实对称矩阵广义特征值反问题: 问题IGEP,给定X∈R~(n×m),1=diag(λ_II_k_I,…,λ_pI_k_p)∈R~(n×m),并且λ_I,…,λ_p互异,sum from i=1 to p(k_i=m,求K,M∈SR~(n×n),或K∈SR~(n×n),M∈SR_0~(n×m),或K,M∈SR_0~(n×n),或K∈SR~(n×n),M∈SR_+~(n×n),或K∈SR_0~(n×n),M∈SR_+~(n×n),或K,M∈SR_+~(n×m), (Ⅰ)使得 KX=MXA, (Ⅱ)使得 X~TMX=I_m,KX=MXA,其中SR~(n×n)={A∈R~(n×n)|A~T=A},SR_0~(n×n)={A∈SR~(n×n)|X~TAX≥0,X∈R~n},SR_+~(n×n)={A∈SR~(n×n)|X~TAX>0,X∈R~n,X≠0}. 利用矩阵X的奇异值分解和正交三角分解,我们给出了上述问题的解的表达式.  相似文献   

14.
反对称正交对称矩阵反问题   总被引:6,自引:0,他引:6  
周富照  胡锡炎 《数学杂志》2005,25(2):179-184
本文讨论一类反对称正交对称矩阵反问题及其最佳逼近.研究了这类矩阵的一些性质,利用这些性质给出了反问题解存在的一些条件和解的一般表达式,不仅证明了最佳逼近解的存在唯一性,而且给出了此解的具体表达式.  相似文献   

15.
非齐次对称特征值问题   总被引:5,自引:0,他引:5  
引言 用SR~(n×n)表示所有。n×n实对称矩阵的集合。R~n表示n维线性空间。||·||_2表示向量的Euclid范数或矩阵的谱范数。 本文研究如下问题: 问题ISEP 给定矩阵A∈SR~n×n和向量b∈R~n,求实数λ和向量X∈R~n使得 AX=λX+b, (1) ||X||_2=1. (2) 若b=0,则问题ISEP就是通常的实对称矩阵特征值问题,若b≠0,则问题ISEP称为非齐次对称特征值问题,使(1)和(2)式成立的数λ和向量X分别称为非齐次特征值和相应的非齐  相似文献   

16.
Derivatives of eigenvalues and eigenvectors with respect to parameters in symmetric quadratic eigenvalue problem are studied. The first and second order derivatives of eigenpairs are given. The derivatives are calculated in terms of the eigenvalues and eigenvectors of the quadratic eigenvalue problem, and the use of state space representation is avoided, hence the cost of computation is greatly reduced. The efficiency of the presented method is demonstrated by considering a spring-mass-damper system.  相似文献   

17.
Nonlinear rank-one modification of the symmetric eigenvalue problem arises from eigenvibrations of mechanical structures with elastically attached loads and calculation of the propagation modes in optical fiber. In this paper, we first study the existence and uniqueness of eigenvalues, and then investigate three numerical algorithms, namely Picard iteration, nonlinear Rayleigh quotient iteration and successive linear approximation method (SLAM). The global convergence of the SLAM is proven under some mild assumptions. Numerical examples illustrate that the SLAM is the most robust method.  相似文献   

18.
张玉海 《计算数学》2001,23(3):333-342
1.引言 设A(c)=(aij(c))是n阶实矩阵,其元素aij(c)(i,j=1,…,n)是参变量c=(C1,…,cn)T的实解析函数,λ1(c),…,λn(C)是矩阵A(c)的特征值,λ1,…,λn是给定的实数,代数特征值反问题[4]就是研究如何求解实的c,使A(c)的特征值为给定的λ1,…,λn. 假设给定的n个数λ1,…,λn互异,且问题的解存在(解不存在时可考虑某种形式的最小二乘解),过去的研究一般是直接研究或将问题转化为如下等价的非线性方程组 det(A(c卜人I)一0, i= 1,…,…  相似文献   

19.
§1 IntroductionWe considerthe following inverse eigenvalue problem offinding an n-by-n matrix A∈S such thatAxi =λixi,i =1,2 ,...,m,where S is a given set of n-by-n matrices,x1 ,...,xm(m≤n) are given n-vectors andλ1 ,...,λmare given constants.Let X=(x1 ,...,xm) ,Λ=(λ1 ,λ2 ,...,λm) ,then the above inverse eigenvalue problemcan be written as followsProblem Given X∈Cn×m,Λ=(λ1 ,...,λm) ,find A∈S such thatAX =XΛ,where S is a given matrix set.We also discuss the so-called opti…  相似文献   

20.
实对称矩阵的两类逆特征值问题   总被引:95,自引:11,他引:84  
孙继广 《计算数学》1988,10(3):282-290
§gi.两类逆特征值问题先说明一些记号.R~(m×n)是所有m×n实矩阵的全体,R~n=R~(n×1),R=R~1;SR~(n×n)是 所有n×n实对称矩阵的全体;OR~(n×n)是所有n×n实正交矩阵的全体;I~((n))是n阶单位矩阵;A~T是矩阵A的转置;A>0表示A是正定的实对称矩阵.?(A)是矩阵A的列空间;A~+是矩阵A的Moore-Penrose广义逆;P_A=AA~+表示到?(A)的正交投影.λ(A)是A的特征值的全体;λ(K,M)是广义特征值问题K_x=λM_x的特征值的  相似文献   

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