Inverse Eigenvalue Problems For Jacobi And Unitary Hessenberg Matrices

Structured inverse eigenvalue problems(SIEP) arise in a variety of applications. For example, solid mechanics, particle physics, mechanism design, system identification and so on. Generally speaking, the research of SIEP is concentrated on the following problems: the theory of solvability (necessary or/and sufficient conditions), the practice of computability(existence, uniqueness and stability), the analysis of sensitivity and the reality of applicability.In this paper, two kinds of structure inverse eigenvalue problems are discussed. Firstly, an inverse eigenvalue problem for Jacobi matrices is presented: we could construct the Jacobi matrix T if we know the spectral data: the eigenvalues of T and the eigenvalues of T₁ and of T₂, where T₁ is different to the k × k leading principal submatrix of T only at the (k, k) position, while T₂ is different to the (n - k) × (n - k) remain submatrix only at the (1, 1) position. And the sufficient and necessary conditions are obtained and the uniqueness of T is discussed and an algorithm for solving the inverse problem is provided. The other kind of structure inverse eigenvalue problem is for unitary Hessenberg matrices with positive subdiagonal elements. That is, a unitary Hessenberg matrices with positive sub-diagonal elements can be constructed when its eigenvalues and the eigenvalues of H₁₁ and H₂₂ are known. Here H₁₁ and H₂₂ are rank-one modifications of k × k leading principal submatrix of H and of its (n - k) × (n - k) remain submatrix respectively. In the end, the uniqueness of H and an algorithm is obtained.In this paper, we put forward a new kind of inverse problem. And we make use of the divide and conquer method for the eigenvalue problems to divide the matrix into two partitions. By discussing the separate relations between these given eigenvalues, we construct the eigenvectors from the given eigenvalues and present relevant algorithms for solving these inverse problems.