Several Kinds Of Matrix Inverse Eigenvalue Problem

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. Problemâ… is to construct A_n by the minimal and maximaleigenvalues of its all leading principal submatrices A_j(j = 1,2,…,n).Problemâ…¡is 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 (λ₁,μ₁)and (λ₂,μ₂),two nonzero real vectors x,y,and a diagonal matrix D =diag(d₁,d₂,…,d_n)(d_i≠0,i=1,2,…,n),find n×n Jacobi matrices A,B,such that ((λ₁,μ₁),x) and ((λ₂,μ₂),y) are the eigenpairs of the coupled general- ized eigenvalue problemand D^-1 A,D^-1B 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 numberσ₁,σ₂,.…,σ_n,two nonzero real vectorsx = (x₁,x₂,…,x_m)^T,y = (y₁,y₂,…,y_n)^T,find m×n real matrix A,such thatσ₁,σ₂,…,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.