On Substochastic Inverse Eigenvalue Problems with the Corresponding Eigenvector Constraints

SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1689-1719, September 2024.
Abstract. We consider the inverse eigenvalue problem of constructing a substochastic matrix from the given spectrum parameters with the corresponding eigenvector constraints. This substochastic inverse eigenvalue problem (SstIEP) with the specific eigenvector constraints is formulated into a nonconvex optimization problem (NcOP). The solvability for SstIEP with the specific eigenvector constraints is equivalent to identifying the attainability of a zero optimal value for the formulated NcOP. When the optimal objective value is zero, the corresponding optimal solution to the formulated NcOP is just the substochastic matrix that we wish to construct. We develop the alternating minimization algorithm to solve the formulated NcOP, and its convergence is established by developing a novel method to obtain the boundedness of the optimal solution. Some numerical experiments are conducted to demonstrate the efficiency of the proposed method.