On Inverse Toeplitz Eigenvalue Problem

Toeplitz matrices play an important role in many practical applications such as signal processing, system identification and image processing. This paper is concerned with the inverse eigenvalue problems for symmetric Toeplitz matrices. A kind of inverse problem for constructing a real symmetric Toeplitz matrix from the given k eigenpairs is proposed. By using the special structure of symmetric Toeplitz matrices, the Kronecker product and the vec operator of matrices, the problem is transformed into the system of linear equations. Some necessary and sufficient conditions for the solvability of the problem are given. The general solutions of the problem are presented.This paper also considers another kind of inverse problem for constructing a real symmetric Toeplitz matrix from the given n real numbers as its eigenvalues. The Newton method is one of the important methods for solving the problem. However every step of the Newton iteration requires the solutions of large nonsymmetric linear systems. In this paper we use iterative methods (inner iterations) for solving these systems approximately, and give the inexact Newton method for solving large inverse Toeplitz eigenvalue problem. This method can avoid the oversolving problem of Newton method and hence improve the efficiency. The convergence of the inexact Newton method is also analyzed in this paper. The numerical results show that the inexact Newton method is better than the Newton method.