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In this paper, we consider the inverse eigenvalue problem for the positivedoubly stochastic matrices, which aims to construct a positive doublystochastic matrix from the prescribed realizable spectral data. By using thereal Schur decomposition, the inverse problem is written as a nonlinear matrixequation on a matrix product manifold. We propose monotone and nonmonotoneRiemannian inexact Newton-CG methods for solving the nonlinear matrix equation.The global and quadratic convergence of the proposed methods is establishedunder some assumptions. We also provide invariant subspaces of the constructedsolution 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.

Riemannian Newton-CG Methods for Constructing a Positive Doubly Stochastic Matrix From Spectral Data