  In this paper, we consider the inverse eigenvalue problem for the positive
doubly stochastic matrices, which aims to construct a positive doubly
stochastic matrix from the prescribed realizable spectral data. By using the
real Schur decomposition, the inverse problem is written as a nonlinear matrix
equation on a matrix product manifold. We propose monotone and nonmonotone
Riemannian inexact Newton-CG methods for solving the nonlinear matrix equation.
The global and quadratic convergence of the proposed methods is established
under some assumptions. We also provide invariant subspaces of the constructed
solution to the inverse problem based on the computed real Schur decomposition.
Finally, we report some numerical tests, including an application in digraph,
to illustrate the effectiveness of the proposed methods.
