Numerical Methods for Solving Inverse Eigenvalue Problems for Nonnegative Matrices

Abstract

Presented here are two related numerical methods, one for the inverse eigenvalue problem for nonnegative or stochastic matrices and another for the inverse eigenvalue problem for symmetric nonnegative matrices. The methods are iterative in nature and utilize alternating projection ideas. For the algorithm for the symmetric problem, the main computational component of each iteration is an eigenvalue-eigenvector decomposition, while for the other algorithm, it is a Schur matrix decomposition. Convergence properties of the algorithms are investigated and numerical results are also presented. While the paper deals with two specific types of inverse eigenvalue problems, the ideas presented here should be applicable to many other inverse eigenvalue problems, including those involving nonsymmetric matrices.