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Inverse Eigenvalue Problem broad range of applications, for example, discrete inverse problems of mathematical physics, control design, system parameter identification, seismic tomography technology, remote sensing techniques, principal component analysis, the antenna signal processing, geophysics, molecular spectroscopy, structural analysis, circuit theory, mechanical system simulation and many other areas. Matrix inverse eigenvalue problem is an important branch of numerical algebra. Matrix Inverse Eigenvalue Problem of content: given characteristic values ??or characteristics of the constructed out of a particular class required to meet certain spectrum constraint matrix and the best approximation. This paper discusses the proposed reflexive inverse matrix eigenvalue problem, mainly in the following aspects:? Inverse eigenvalue problem, ie given matrix X, the diagonal matrix Λ and non-singular Hermite matrix P, seeking to be reflexive matrix A such that AX = XΛ. We give necessary and sufficient conditions for the solvability of the problem and the solution of the general expression. We intend to get these issues all reflexive matrices is denoted by S X, Λ . Then discuss the best approximation problem: given an arbitrary matrix A * , seeking S X, Λ in a matrix norm can meet with A * < / sup> The best approximation, we show that this problem has a unique solution, and gives expression to understand, and the corresponding algorithms and numerical examples. By category compared reflexive inverse matrix eigenvalue problem, using similar method to study the proposed anti-reflexive matrix inverse eigenvalue problems, and gives expression to understand, and the corresponding algorithms and numerical examples. Left and right inverse eigenvalue problem, that is a given matrix X, Y, the diagonal matrix Λ, Γ and non-singular Hermite matrix P, seeking to be reflexive matrix A such that AX = XΛ, YHA = ΓYH. Discussion to understand the necessary and sufficient conditions exist and gives the A * and the general expression of the general expression of the best approximation and numerical experiments examples. Generalized inverse eigenvalue problem, ie given matrix X, the diagonal matrix Λ and non-singular Hermite matrix P, seeking to be reflexive matrix A, B such that AX = BXΛ. Find A, B and the general expression to be obtained in the above problems all reflexive array denoted S A, B . Then discuss the best approximation for this problem: For any given matrix A * , B * , Matrix A, B ∈ SA, B, makes Fan number sense A, B are A * , B * The best approximation, we prove that this problem has a unique solution and the solution is given expression, and the corresponding Examples of algorithms and numerical experiments.
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