A Numerical Approach to the Inverse Toeplitz Eigenproblem
Dirk Laurie · SIAM Journal on Scientific and Statistical Computing · 1988
A Toeplitz matrix $T = {\operatorname{Toep}}(t_0 \cdots t_{n - 1} ) = [t_{ij} ]$ is a matrix with the property that $t_{ij} = t_{i - j} $, i, $j = 1, \cdots ,n$. The inverse Toeplitz eigenproblem is: construct a real symmetric Toeplitz matrix having a given set of n real numbers as its eigenvalues. This paper presents a numerical approach to the problem. At each stage, a Toeplitz matrix $T_k $ is constructed having the given set of numbers as its Rayleigh quotients with respect to the current matrix $Q_k $ of approximate eigenvectors. The new $Q_{k + 1} $ is found by diagonalizing $T_k $. Convergence could not be proved. Numerical evidence indicates that the algorithm behaves like a typical quadratically convergent algorithm. In the course of testing the algorithm, a counterexample was found to the conjecture of Delsarte and Genin that the eigenvectors of a symmetric Toeplitz matrix, corresponding to eigenvalues arranged in decreasing order, alternate between reciprocal and antireciprocal vectors.