A Riemannian Fletcher--Reeves Conjugate Gradient Method for Doubly Stochastic Inverse Eigenvalue Problems

Abstract

We consider the inverse eigenvalue problem of reconstructing a doubly stochastic matrix from the given spectrum data. We reformulate this inverse problem as a constrained nonlinear least squares problem over several matrix manifolds, which minimizes the distance between isospectral matrices and doubly stochastic matrices. Then a Riemannian Fletcher--Reeves conjugate gradient method is proposed for solving the constrained nonlinear least squares problem, and its global convergence is established. An extra gain is that a new Riemannian isospectral flow method is obtained. Our method is also extended to the case of prescribed entries. Finally, some numerical tests are reported to illustrate the efficiency of the proposed method.