Several Classes Of Constrained Matrix Inverse Eigenproblem And Associated Optimal Approximation Problem

A matrix inverse eigenvalue problem concerns the reconstruction of a matrix from prescribed spectral data. The spectral data involved may consist of the complete or only partial information of eigenvalues or eigenvectors. The solution to a matrix inverse eigenvalue problem should satisfy two constraints—the spectral constraint referring to the prescribed spectral data and the structural constraint referring to the desirable structure. When the constraint conditions are different, we can get a variety of inverse eigenvalue problems.This Ph.D. thesis considers the following two problems.Problem Ⅰ (Constrained Inverse Eigenproblem) Given a subset S of all n × n matrices, and a real number α > 0, and given matrices Z ∈ F^n×k (k < n) and Λ ∈F^k×k, where F denote real number field or complex number field, find a subset φ(Z, Λ) (?) S such thatand to find the subset φ(Z,Λ,α) (?) φ(Z, Λ) such that all of the remaining eigenvalues of any matrix in φ(Z,Λ,α) are located in the given closed disk D_α = {z||z}≤ α,z∈C}.Problem Ⅱ (Optimal Approximation Problem) Given a matrix B ∈ R^n×n, find A_B ∈ φ(Z,Λ,α) such thatwhere ||·|| is the Frobenius norm.The main achievements of this dissertation are as follows.1. When F = C and S is the set of real n × n antisymmetric matrices, we first present the equivalent formulation of problem I in real number field, then the sufficient and necessary conditions of the solvability for problem I are obtained, and the expressions of solutions of problems I and II are provided.2. When F = R and S is the set of all n × n bisymmetric matrices, the given closed disk D_α in problem I is replaced by a given interval [α,β], where α and β are two given real numbers and α<β. Based on the special structure and special spectral properties of bisymmetric matrices, problems I and II are essentially decomposed into the same kind subproblems for two real symmetric matrices with smaller dimensions, the solutions of problems I and II are obtained,and the numerical algorithm and two examples to solve problem II are also given. When S is the set of all n x n symmetric reflexive matrices we have also discussed the above two problems similarly.3. When F = R and S is the set of all n x n symmetric and anti-persymmetric matrices, we present reasonably the mathematical description of problem I based on the spectral property of this kind of matrices. By using the special structure and special spectral properties of symmetric and anti-persymmetric matrices, problems I and II are essentially deduced to the same kind problems for real nxn symmetric matrices with special structure, and then the solutions of problems I and II are obtained, and the numerical algorithm and two examples to solve problem II are also given. When S is the set of all n x n symmetric antireflexive matrices we have also discussed the above two problems similarly.4. When F = C and S is the set of all n x n anti-bisymmetric matrices, we present reasonably the mathematical description of problem I in complex number field. Based on the special structure and special spectral properties of anti-bisymmetric matrices, problems I and II are essentially decomposed into the same kind subproblems for real antisymmetric matrices with smaller dimensions, we present the same kind subproblems for real antisymmetric matrices in real number field, and then we present the corresponding problems Io and Ho which are the formulations of problems I and II in real number field, and the solutions of problems Io and II0 are provided. We also present the numerical algorithm and two examples to solve problem Ho. When S is the set of all n x n antisymmetric reflexive matrices we have also discussed the above two problems similarly.5. When F — C and S is the set of all n x n antisymmetric and persymmetric matrices, we present the formulation of problem I in complex number field. By using the special structure and special spectral properties of antisymmetric and persymmetric matrices, problems I and II are essentially deduced to the same kind problems for real nxn antisymmetric matrices with special structure, we present the same kind problems for the special antisymmetric matrices in real number field, and then we present the corresponding problems Io and II0 which are the formulations of problems I and II in real number field, the solutions of problems Io and Ho are obtained. And the numerical algorithm and two examples to solve problem Ho are given.This Ph.D. thesis is supported by the National Natural Science Foundation of China.