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.

翻译：标准多参数特征值问题（MEPs）是由$k\ge 2$个线性$k$参数方阵束构成的方程组。近年来，一种新型多参数特征值问题应运而生：矩形MEP（RMEP），它仅涉及一个多元矩形矩阵束，目标在于寻找使得该矩阵束秩亏损的参数组合。此类问题的应用包括求解最优最小二乘自回归滑动平均（ARMA）模型以及最优最小二乘实现的自洽线性时不变（LTI）动力系统。针对线性与多项式RMEPs，我们给出了解的个数，并展示了如何通过转化为标准MEP进行数值求解。为此转化过程，我们提出了具有特定单项式结构的二次多元矩阵多项式的新线性化方法，同时考虑了矩形与方形多元矩阵多项式构成的混合系统。该数值方法在计算效率上显著优于当前唯一可用的多项式RMEP数值方法——块Macaulay方法。