Solving the inverse eigenvalue problem via the eigenvector matrix

Abstract

A numerical algorithm for the inverse eigenvalue problem for symmetric matrices is developed, based on continually updating the eigenvector matrix using plane rotations. A fundamental tool in the algorithm is a matrix formed from the Rayleigh quotients of the eigenvectors with respect to each of the basis matrices involved, with the basis orthonormalized with respect to the Frobenius inner product. Two criteria of closeness to a solution are defined, either of which makes it possible to monitor progress towards a solution. The computational questions involved in this approach are examined in detail. Numerical examples are given for Toeplitz and centrosymmetric tridiagonal matrices. The new algorithm is much more robust than Newton's method.