  Standard multiparameter eigenvalue problems (MEPs) are systems of $k\ge 2$
linear $k$-parameter square matrix pencils. Recently, a new form of
multiparameter eigenvalue problems has emerged: a rectangular MEP (RMEP) with
only one multivariate rectangular matrix pencil, where we are looking for
combinations of the parameters for which the rank of the pencil is not full.
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. 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. For the transformation we
provide new linearizations for quadratic multivariate matrix polynomials with a
specific structure of monomials and consider mixed systems of rectangular and
square multivariate matrix polynomials. This numerical approach seems
computationally considerably more attractive than the block Macaulay method,
the only other currently available numerical method for polynomial RMEPs.
