The multiparameter matrix pencil problem (MPP) is a generalization of the one-parameter MPP: given a set of $m\times n$ complex matrices $A_0,\ldots, A_r$, with $m\ge n+r-1$, it is required to find all complex scalars $\lambda_0,\ldots,\lambda_r$, not all zero, such that the matrix pencil $A(\lambda)=\sum_{i=0}^r\lambda_iA_i$ loses column rank and the corresponding nonzero complex vector $x$ such that $A(\lambda)x=0$. This problem is related to the well-known multiparameter eigenvalue problem except that there is only one pencil and, crucially, the matrices are not necessarily square. In this paper, we give a full solution to the two-parameter MPP. Firstly, an inflation process is implemented to show that the two-parameter MPP is equivalent to a set of three $m^2\times n^2$ simultaneous one-parameter MPPs. These problems are given in terms of Kronecker commutator operators (involving the original matrices) which exhibit several symmetries. These symmetries are analysed and are then used to deflate the dimensions of the one-parameter MPPs to $\frac{m(m-1)}{2}\times\frac{n(n+1)}{2}$ thus simplifying their numerical solution. In the case that $m=n+1$ it is shown that the two-parameter MPP has at least one solution and generically $\frac{n(n+1)}{2}$ solutions and furthermore that, under a rank assumption, the Kronecker determinant operators satisfy a commutativity property. This is then used to show that the two-parameter MPP is equivalent to a set of three simultaneous eigenvalue problems. A general solution algorithm is presented and numerical examples are given to outline the procedure of the proposed algorithm.

翻译：多参数矩阵束问题（MPP）是单参数MPP的推广：给定一组$m\times n$复矩阵$A_0,\ldots, A_r$（其中$m\ge n+r-1$），需要找出所有不全为零的复数标量$\lambda_0,\ldots,\lambda_r$，使得矩阵束$A(\lambda)=\sum_{i=0}^r\lambda_iA_i$的列秩降低，并确定对应的非零复向量$x$满足$A(\lambda)x=0$。该问题与著名的多参数特征值问题相关，但区别在于仅存在一个矩阵束，且关键之处在于矩阵不一定是方阵。本文对双参数MPP给出了完整解法。首先，通过膨胀化处理证明双参数MPP等价于三个$m^2\times n^2$维联立单参数MPP的集合。这些问题通过克罗内克换位子算子（涉及原始矩阵）表述，这些算子呈现多种对称性。本文分析了这些对称性，并利用其将单参数MPP的维度压缩至$\frac{m(m-1)}{2}\times\frac{n(n+1)}{2}$，从而简化了数值求解过程。当$m=n+1$时，研究证明双参数MPP至少存在一个解，且通常存在$\frac{n(n+1)}{2}$个解；此外，在秩假设条件下，克罗内克行列式算子满足交换性。基于此，本文进一步证明双参数MPP等价于三个联立特征值问题的集合。文中提出了通用求解算法，并通过数值算例阐述了所提算法的实施流程。