A new constrained optimization model for solving the nonsymmetric stochastic inverse eigenvalue problem.

The stochastic inverse eigenvalue problem aims to reconstruct a stochastic matrix from its spectrum. Recently, Zhao et al. [A geometric nonlinear conjugate gradient method for stochastic inverse eigenvalue problems. SIAM J Numer Anal. 2016;54(4):2015–2035] proposed a constrained optimization model on the manifold of so-called isospectral matrices and adapted a modified. Polak-Ribière-Polyak conjugate gradient method to the geometry of this manifold. However, not every stochastic matrix is an isospectral one and the model in Zhao et al. is based on the assumption that for each stochastic matrix there exists a (possibly different) isospectral, stochastic matrix with the same spectrum. We are not aware of such a result in the literature, but prove the claim at least for 3 × 3 matrices. In this paper, we suggest to extend the above model by considering matrices which differ from isospectral ones only by multiplication with a block diagonal matrix with 2 × 2 blocks from the special linear group. First, we show that each stochastic matrix can be written in such a form. We prove that our model has a minimizer and show how the Polak–Ribiére–Polyak conjugate gradient method works on the corresponding more general manifold. We demonstrate by numerical examples that the new, more general method performs similarly as those in Zhao et al. [ABSTRACT FROM AUTHOR]