On the solution of rectangular multiparameter eigenvalue problems

Standard multiparameter eigenvalue problems (MEPs) are systems of $k\ge 2$ linear $k$-parameter square matrix pencils. Recently, a new form has emerged, a rectangular MEP (RMEP), with only one multivariate rectangular matrix pencil, where we are looking for combinations of the parameters where the rank of the pencil is not full. For linear and polynomial RMEPs we give the number of solutions and show how these problems can be solved numerically by a transformation into a standard MEP. Applications include finding the optimal least squares autoregressive moving average (ARMA) model and the optimal least squares realization of autonomous linear time-invariant (LTI) dynamical system. The new numerical approach seems computationally considerably more attractive than the block Macaulay method, the only other currently available numerical method for polynomial RMEPs.