On the Two-Parameter Matrix Pencil Problem

SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1318-1340, September 2024.
Abstract. The multiparameter matrix pencil problem (MPP) is a generalization of the one-parameter MPP: Given a set of [math], [math] complex matrices [math] with [math], it is required to find all complex scalars [math], not all zero, such that the matrix pencil [math] loses column rank and the corresponding nonzero complex vector [math] such that [math]. We call the [math]-tuple [math] an eigenvalue and the corresponding vector [math] an eigenvector. 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. This paper uses our preliminary investigation in F. F. Alsubaie [[math] Optimal Model Reduction for Linear Dynamic Systems and the Solution of Multiparameter Matrix Pencil Problems, PhD thesis, Imperial College London, 2019], which presents a theoretical study of the multiparameter MPP and its applications in the [math] optimal model reduction problem, to give a full solution to the two-parameter MPP. First, an inflation process is implemented to show that the two-parameter MPP is equivalent to a set of three [math] simultaneous one-parameter MPPs. These problems are given in terms of Kronecker commutator operators (involving the original matrices) that exhibit several symmetries. These symmetries are analyzed and are then used to deflate the dimensions of the one-parameter MPPs to [math], thus simplifying their numerical solution. In the case in which [math], it is shown that the two-parameter MPP has at least one solution and generically [math] 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 of dimension [math]. A general solution algorithm is presented and numerical examples are given to outline the procedure of the proposed algorithm.