Isospectral flows and the inverse eigenvalue problem for Toeplitz matrices

摘要

The inverse eigenvalue problem for Toeplitz matrices (ITEP), concerning the reconstruction of a symmetric Toeplitz matrix from prescribed spectral data, is considered. To numerically construct such a matrix the approach introduced by Chu in (SIAM Rev. 40(1) (1998) 1–39) is followed. He proposed to solve the ITEP by using an isospectral flow whose equilibria are symmetric Toeplitz matrices. In this paper we study the previous isospectral flow for reversed times and we obtain some formal properties of the solution. The case n=3 for ITEP is analytically investigated by following an approach different from the one in (Chu, SIAM Rev. 40(1) (1998) 1–39). We prove that the flow globally converges to a regular Toeplitz matrix starting from a tridiagonal symmetric and centro-symmetric matrix. Numerical experiments confirm the above results and suggest their extension in higher dimension.