Inverse Eigenvalue Problem For Reflexive Matrices

The inverse eigenvalue problems(IEP)for matrices are studied in manyfields. They arise in a remarkable variety of applications. The list includes dis-persed mathematical physical inverse problem, control design, system parameteridentification, 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. There are also some inverse eigenvalue problems for matrices in numericalalgebra. An inverse eigenvalue problem is concentrated on the following problem:given eigenvalue and eigenpairs, whether or not we can construct the specificmatrix or the optimal approximation of a matrix under given spectral restriction.In this paper, we mainly discuss the inverse eigenvalue problems for re?exivematrices. In chapter 1, we give some preliminary knowledge and lemmas aboutthis paper. In chapter 2, we first consider the inverse eigenvalue problem as fol-lows: given a matrix X, diagonal matrixΛand a generalized re?ection matrixP, find a re?exive matrix solutions A of AX = XΛ. The su?cient and necessaryconditions are obtained, and a general representation of such a matrix is pre-sented. We denote the set of such matrices by S_A. Then the best approximationproblem for the inverse problem is discussed, that is: given an arbitrary A^* , finda matrix A|^∈S_A, which is nearest to A^* in the Frobenius norm. We show that thebest approximation is unique and provide an expression for this nearest matrix,we also give the stability analysis and numerical example for this problem. Inchapter 3, we study the inverse generalized eigenvalue problem as follows: given amatrix X, diagonal matrixΛand a generalized reflection matrix P, the reflexivematrix solutions A, B of AX = BXΛare considered. The general representationof such a solution is presented. We denote the set of such matrices by S_AB. Thenthe optimal approximation problem is discussed, that is: given arbitrary matri-ces A^*, B^*, find matrix A|^,B|^∈S_AB which is nearest to (A^*,B^*) in the Frobenius...