The Research On Some Problems For Hermitian Reflexive Matrix

A matrix inverse eigenvalue problem concerns the reconstruction of a matrix from prescribed spectral date. The spectral data involved may consist of the complete or only partial information of eigenvalue or eigenvectors. The inverse eigenvalue problem is mainly used in control design, geophysics, molecular spectroscopy, particle physics, structure analysis and so on. The constrained matrix equation problem is to find the solution of a matrix equation in a constrained matrix set, which is applied in structural design, system identification, automatic control, vibration theory, computational physics, nonlinear programming, and civil engineering and so on. The inverse eigenvalue and matrix equation problem of Hermitian reflexive matrix is widely used in practice due to the special nature. So, in this paper, the inverse eigenvalue problem and the constraint matrix equation problem are considered, which mathematical description is as following,ProblemⅠ. Given X, Y∈Cn×m,Λ=diag (μ1,…,μm),Γ=(τ1,…,τm)∈Rm×m, find A∈HCrn×n(P),such that AX=XA, YH A=ΓYHProblemⅡ. Given X∈Cn×m,Λ=diag (μ1,…,μm)∈Cm×m, find A, B∈HCrn×n (P),such that AX=BXΛProblemⅢ. Given A,B,C∈Cn×m, find X, Y∈HCrn×n (P), such that‖AX+BY-C‖=minProblemⅣ. Given X, B∈Cn×m, find A∈HCrn×n (P), such that Where the S is a linear manifold.ProblemⅤ. For any given matrices A*, B*∈Cn×n,find A∈T,B∈T, such that Or Where the T is the solutions of the above problemⅠ,Ⅱ,ⅢandⅣrepectively.The main research content of this paper are as following:(1) By using the property of the eigenvalue and the eigenvector of the Hermitian matrix, the mathematical description on inverse eigenvalue problem is presented. The property and structure of the hermitian reflexive matrix are considered, and the denotative theorem of the hermitian reflexive matrix is derived.(2) By using the denotative theorem which is obtained above of hermitian reflexive matrix and the SVD decompounding method, the sufficient and necessary conditions and the expression of the general solutions for problemⅠandⅣare obtained. At the same time, by using the straightened matrix, the expressions of the general solutions of the problemⅡandⅢare obtained respectively.(3) Based on the 2, furthermore, the least-squares solutions of the above problem are considered, and the general expression of the solutions are presented.(4) Corresponding the problemⅤ, the existence and uniqueness of the optimal approximation solutions are proved respectively and the expressions of the best optimal approximation solutions are derived.(5) Finally, in order to prove the correctness of the method and the solutions, the numerical algorithms and the numerical example for each problem are given respectively.The dissertation has gained the support from the National Natural Science Foundation of China.