Study Of Inverse Eigenvalue Problems For Symmetric Nonnegative Matrices By Isospectral Methods

Inverse eigenvalue problem,that is,constructing a matrix that meets the conditions according to the given eigenvalues and properties of the matrix.There are many types of inverse eigenvalue problems,whose difficulty depends on the structure of the matrix to be reconstructed and the available eigenvalue information.The typical ones include inverse eigenvalue problems of nonnegative matrices and inverse eigenvalue problems of Jacobi matrices.Such problems have applications in cybernetics,mechanical system simulation,geophysics,molecular spectroscopy,structural analysis,string vibration,circuit theory,and graph theory.For the inverse eigenvalue problem,some scholars transform it into an optimization problem,and then use the existing ODE solver or choose the appropriate optimization method according to different problems.Some scholars write it in the form of isospectral flow and then use differential equation to solve it.Isospectral flow is widely used in many fields such as molecular dynamics,biology,micromagnetism,linear algebra and solid state physics.Many interesting problems can be written in the form of isospectral flow,and the inverse eigenvalue problem is one of the most important ones.In this thesis,the isospectral method is used to solve the inverse eigenvalue problem of symmetric nonnegative matrices.The problem is transformed into an optimization problem,and then into an isospectral flow problem.Considering the need to maintain the isospectral property of the problem,several kinds of numerical algorithms which can preserve eigenvalue and symmetry are constructed by using the idea of structure-preserving algorithm.The simulation results under different eigenvalues are given by numerical experiments,and compared with the non-structure-preserving algorithm,the effectiveness of the proposed algorithm is verified.