On the Two-Parameter Matrix Pencil Problem
S. K. Gungah, Fawwaz F. Alsubaie, Imad M. Jaimoukha · SIAM Journal on Matrix Analysis and Applications · 2024
Abstract. The multiparameter matrix pencil problem (MPP) is a generalization of the one-parameter MPP: Given a set of [Formula: see text], [Formula: see text] complex matrices [Formula: see text] with [Formula: see text], it is required to find all complex scalars [Formula: see text], not all zero, such that the matrix pencil [Formula: see text] loses column rank and the corresponding nonzero complex vector [Formula: see text] such that [Formula: see text]. We call the [Formula: see text]-tuple [Formula: see text] an eigenvalue and the corresponding vector [Formula: see text] 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 [[Formula: see text] 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 [Formula: see text] 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 [Formula: see text] 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 [Formula: see text], thus simplifying their numerical solution. In the case in which [Formula: see text], it is shown that the two-parameter MPP has at least one solution and generically [Formula: see text] 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 [Formula: see text]. A general solution algorithm is presented and numerical examples are given to outline the procedure of the proposed algorithm.