A Riemannian Optimization Approach for Solving the Generalized Eigenvalue Problem for Nonsquare Matrix Pencils

In this paper, based on the Riemannian optimization approach we propose a Riemannian nonlinear conjugate gradient method with nonmonotone line search technique for solving the l parameterized original problem on generalized eigenvalue problems for nonsquare matrix pencils, which was first proposed by Chu and Golub (SIAM J Matrix Anal Appl 28:770–787, 2006). The new innovative approach is to reformulate the original optimization problem as a feasible optimization problem over a certain real product manifold. The global convergence of the proposed method is then established. Some numerical tests are given to demonstrate the efficiency of the proposed method. Comparisons with some latest methods are also given.