Inverse Eigenvalue Problems For Quasi Tridiagonal Matrices

The main content of this paper is to investigate inverse eigenvalue problem for a class of complex symmetric matrix, that is, quasi tridiagonal matrix. We expand non-zero elements of the matrix in the number field, from real domain whose theory relatively mature to complex domain, and get ideal conclusions and a relatively stable algorithm. The following is to sum up the main content of this thesis in three aspects.Firstly, we study the quasi tridiagonal matrix eigenvalue problem. In this thesis,we discuss the eigenvalues of the quasi tridiagonal matrix and principal submatrix,as well as the relationship of them, we have proposed the conclusion that all the eigenvalues of the quasi tridiagonal matrix are real and simple and interlace the eigenvalues of its principal submatrix, which are real and simple as well. In this part,we confirm number field and multiplicity of eigenvalues to make conditions of inverse eigenvalue problem clear, and make the research of inverse eigenvalue problem convenient.Secondly, we discuss inverse eigenvalue problems for quasi tridiagonal matrix.In this part, the claw matrix is constructed, that is, boundary elements of the claw matrix are constructed by the given eigenvalues, and sufficient conditions are derived for existence of reconstruction of the quasi tridiagonal matrix, under the premise of existence for the solution, followed by the reconstruction of the quasi tridiagonal matrix using the theory of unitary similarity.Finally, we present the algorithm of inverse eigenvalue problem of quasi tridiagonal matrix, and gives three typical numerical examples. At the same time, a Matlab program has been written to test this algorithm via a large amount of data verification. It illustrates that the reconstructed matrix through any given data which ensure the existence of the solution is ideal, and the stability of this algorithm is better.