几类矩阵的逆特征值问题
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摘要
矩阵逆特征值问题的研究领域非常广泛，主要来自于离散的数学物理反问题、控制设计、系统参数识别、地震断层成像技术、主成分分析与勘测、遥感技术、天线讯号处理、地球物理、分子光谱、粒子物理、结构分析、电路理论、机械系统模拟等许多应用领域．矩阵逆特征值问题的研究内容是：对给定的特征值或特征对，能否构造出所要求的特定类的矩阵及满足一定谱约束的最佳逼近．
本文主要讨论了几类矩阵的逆特征值问题．全文共分为五章．第一章介绍了矩阵逆特征值问题的来源、研究内容、发展现状、矩阵逆特征值问题的不同的提法以及本文的结构．
第二章讨论了一类具有特殊形式的矩阵A_n的两类逆特征值问题．问题Ⅰ是由A_n的顺序主子阵A_j(j=1，2，…，n)的最小、最大特征值来构造矩阵A_n；问题Ⅱ是由A_n的顺序主子阵A_j(j=1，2，…，n)的所有特征值来构造矩阵A_n．我们分别给出了两类问题有解的充分必要条件，提供了相应的算法和数值例子，并用数值结果表明我们的算法是很有效的．
第三章研究了两个参数的Jacobi矩阵逆特征值问题(IEP2p)：给定两对不同的实数(λ_1，μ_1)，(λ_2，μ_2)；两非零实向量x=(x_1，x_2，…，x_n)~T，y=(y_1，y_2，…，y_n)~T；对角阵D=diag(d_1，d_2，…，d_n)(d_i≠0，i=1，2，…，n)，求n阶Jacobi矩阵A，B，使得((λ_1，μ_1)，x)，((λ_2，μ_2)，y)为广义特征问题的特征对，且D~(-1)A，D~(-1)B可交换．我们得到了IEP2p存在唯一解的充分必要条件，并给出了相应的算法和数值例子．
第四章考虑的是一类逆奇异值问题．给定非负实数σ_1，σ_2，…，σ_n，两非零实向量x=(x_1，x_2，…，x_m)~T，y=(y_1，y_2，…，y_n)~T，求m×n阶实矩阵A，使得σ_1，σ_2，…，σ_n为A的奇异值，并且x，y分别为A的左右奇异向量．我们基于Householder变换和矩阵秩1的修正的方法得到了问题的算法，而且算法比较经济易于并行，同时给出了相应地数值例子．
第五章讨论了次对角元是正数的为酉上Hessenberg矩阵H的逆特征值问题．当k＜n时，H的k阶顺序主子阵H_k不再是酉阵，其特征值在单位圆的内部，修正H_k得到酉上Hessenberg矩阵(?)_k，使得(?)_k的特征值位于单位圆上．然后由(?)_k(k=1，2，…，n)的最小、最大特征值来构造出唯一的H，我们得到了问题有解的充分必要条件，并通过数值例子加以说明．
The inverse eigenvalue problems (IEP) for matrices are studied in manyfields,they arise in a remarkable variety of applications,the list includes dispersedmathematical physical inverse problem,control design,system parameter iden-tification,seismic tomography,principal component analysis,exploration andremote sensing,antenna array processing,geophysics,molecular spectroscopy,particle physics,structural analysis,circuit theory,and mechanical system simu-lation.An inverse eigenvalue problem is concentrated on the following problem:given eigenvalue and eigenpairs,whether or not we can construct the specific ma-trix or the optimal approximation of a matrix under given spectral restriction.
In this paper,we mainly discuss several kinds of matrix inverse eigenvalueproblems.This thesis comprises five chapters.In Chapter 1,we firstly give a briefreview of the background of the inverse eigenvalue problem,then list differentkinds of inverse eigenvalue problem,finally introduce the structure of this paper.
In Chapter 2,we discuss two inverse eigenvalue problems of a special kindof matrices A_n. ProblemⅠis to construct A_n by the minimal and maximaleigenvalues of its all leading principal submatrices A_j(j = 1,2,…,n).ProblemⅡis to construct A_n by all eigenvalues of its all leading principal submatricesA_j(j = 1,2,…,n).The necessary and sufficient conditions for the solvabilityof the two problems are derived,respectively,and results are constructive.Wealso give the corresponding numerical algorithms and examples,numerical resultsshow good efficient of the algorithms.
In Chapter 3,we study the following inverse eigenvalue problem oftwo-parameter (IEP2p):given two pairs of distinct real number (λ_1,μ_1)and (λ_2,μ_2),two nonzero real vectors x,y,and a diagonal matrix D =diag(d_1,d_2,…,d_n)(d_i≠0,i=1,2,…,n),find n×n Jacobi matrices A,B,such that ((λ_1,μ_1),x) and ((λ_2,μ_2),y) are the eigenpairs of the coupled general- ized eigenvalue problemand D~(-1) A,D~(-1)B are commutative.We propose the necessary and sufficientconditions for existence and uniqueness of IEP2p's solution.Furthermore,cot-responding numerical algorithm and example are included.
In Chapter 4,we concern a kind of matrix inverse singular value prob-lem. Given real nonnegative numberσ_1,σ_2,.…,σ_n,two nonzero real vectorsx = (x_1,x_2,…,x_m)~T,y = (y_1,y_2,…,y_n)~T,find m×n real matrix A,such thatσ_1,σ_2,…,σ_n are the singular values of A,and x,y are the left and right singularvectors,respectively.Based on Householder transformation and rank-one updat-ing,we propose a algorithm which is economical and easily to parallel to solvethe inverse singular value problems,we also give the corresponding numericalexample.
In Chapter 5,we consider the inverse eigenvalue of unitary upper Hessen-berg matrix H whose subdiagonal elements are all positive.Let H_k be the k-thleading principal submatrix of H,H_k is not unitary for k＜n,and it's eigen-values are inside the unit circle,we introduce the modified unitary submatrices(?)_k,such that it's eigenvalues are on the unit circle.H is constructed uniquelyif the minimal and maximal eigenvalues of (?)_k(k = 1,2,…,n) are known,(?)_k isthe modified submatrices of H.We give the necessary and sufficient conditionsfor existence and uniqueness of the solution. Numerical experiment is alsopresented to illustrate our results.
本文主要讨论了几类矩阵的逆特征值问题．全文共分为五章．第一章介绍了矩阵逆特征值问题的来源、研究内容、发展现状、矩阵逆特征值问题的不同的提法以及本文的结构．
第二章讨论了一类具有特殊形式的矩阵A_n的两类逆特征值问题．问题Ⅰ是由A_n的顺序主子阵A_j(j=1，2，…，n)的最小、最大特征值来构造矩阵A_n；问题Ⅱ是由A_n的顺序主子阵A_j(j=1，2，…，n)的所有特征值来构造矩阵A_n．我们分别给出了两类问题有解的充分必要条件，提供了相应的算法和数值例子，并用数值结果表明我们的算法是很有效的．
第三章研究了两个参数的Jacobi矩阵逆特征值问题(IEP2p)：给定两对不同的实数(λ_1，μ_1)，(λ_2，μ_2)；两非零实向量x=(x_1，x_2，…，x_n)~T，y=(y_1，y_2，…，y_n)~T；对角阵D=diag(d_1，d_2，…，d_n)(d_i≠0，i=1，2，…，n)，求n阶Jacobi矩阵A，B，使得((λ_1，μ_1)，x)，((λ_2，μ_2)，y)为广义特征问题的特征对，且D~(-1)A，D~(-1)B可交换．我们得到了IEP2p存在唯一解的充分必要条件，并给出了相应的算法和数值例子．
第四章考虑的是一类逆奇异值问题．给定非负实数σ_1，σ_2，…，σ_n，两非零实向量x=(x_1，x_2，…，x_m)~T，y=(y_1，y_2，…，y_n)~T，求m×n阶实矩阵A，使得σ_1，σ_2，…，σ_n为A的奇异值，并且x，y分别为A的左右奇异向量．我们基于Householder变换和矩阵秩1的修正的方法得到了问题的算法，而且算法比较经济易于并行，同时给出了相应地数值例子．
第五章讨论了次对角元是正数的为酉上Hessenberg矩阵H的逆特征值问题．当k＜n时，H的k阶顺序主子阵H_k不再是酉阵，其特征值在单位圆的内部，修正H_k得到酉上Hessenberg矩阵(?)_k，使得(?)_k的特征值位于单位圆上．然后由(?)_k(k=1，2，…，n)的最小、最大特征值来构造出唯一的H，我们得到了问题有解的充分必要条件，并通过数值例子加以说明．
The inverse eigenvalue problems (IEP) for matrices are studied in manyfields,they arise in a remarkable variety of applications,the list includes dispersedmathematical physical inverse problem,control design,system parameter iden-tification,seismic tomography,principal component analysis,exploration andremote sensing,antenna array processing,geophysics,molecular spectroscopy,particle physics,structural analysis,circuit theory,and mechanical system simu-lation.An inverse eigenvalue problem is concentrated on the following problem:given eigenvalue and eigenpairs,whether or not we can construct the specific ma-trix or the optimal approximation of a matrix under given spectral restriction.
In this paper,we mainly discuss several kinds of matrix inverse eigenvalueproblems.This thesis comprises five chapters.In Chapter 1,we firstly give a briefreview of the background of the inverse eigenvalue problem,then list differentkinds of inverse eigenvalue problem,finally introduce the structure of this paper.
In Chapter 2,we discuss two inverse eigenvalue problems of a special kindof matrices A_n. ProblemⅠis to construct A_n by the minimal and maximaleigenvalues of its all leading principal submatrices A_j(j = 1,2,…,n).ProblemⅡis to construct A_n by all eigenvalues of its all leading principal submatricesA_j(j = 1,2,…,n).The necessary and sufficient conditions for the solvabilityof the two problems are derived,respectively,and results are constructive.Wealso give the corresponding numerical algorithms and examples,numerical resultsshow good efficient of the algorithms.
In Chapter 3,we study the following inverse eigenvalue problem oftwo-parameter (IEP2p):given two pairs of distinct real number (λ_1,μ_1)and (λ_2,μ_2),two nonzero real vectors x,y,and a diagonal matrix D =diag(d_1,d_2,…,d_n)(d_i≠0,i=1,2,…,n),find n×n Jacobi matrices A,B,such that ((λ_1,μ_1),x) and ((λ_2,μ_2),y) are the eigenpairs of the coupled general- ized eigenvalue problemand D~(-1) A,D~(-1)B are commutative.We propose the necessary and sufficientconditions for existence and uniqueness of IEP2p's solution.Furthermore,cot-responding numerical algorithm and example are included.
In Chapter 4,we concern a kind of matrix inverse singular value prob-lem. Given real nonnegative numberσ_1,σ_2,.…,σ_n,two nonzero real vectorsx = (x_1,x_2,…,x_m)~T,y = (y_1,y_2,…,y_n)~T,find m×n real matrix A,such thatσ_1,σ_2,…,σ_n are the singular values of A,and x,y are the left and right singularvectors,respectively.Based on Householder transformation and rank-one updat-ing,we propose a algorithm which is economical and easily to parallel to solvethe inverse singular value problems,we also give the corresponding numericalexample.
In Chapter 5,we consider the inverse eigenvalue of unitary upper Hessen-berg matrix H whose subdiagonal elements are all positive.Let H_k be the k-thleading principal submatrix of H,H_k is not unitary for k＜n,and it's eigen-values are inside the unit circle,we introduce the modified unitary submatrices(?)_k,such that it's eigenvalues are on the unit circle.H is constructed uniquelyif the minimal and maximal eigenvalues of (?)_k(k = 1,2,…,n) are known,(?)_k isthe modified submatrices of H.We give the necessary and sufficient conditionsfor existence and uniqueness of the solution. Numerical experiment is alsopresented to illustrate our results.
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