The multiparameter matrix pencil problem (MPP) is a generalization of the one-
parameter MPP: given a set of r + 1, m × n complex matrices A0, . . . , Ar , with m ≥ n + r − 1, it
is required to find all complex scalars λ0, . . . , λr , not all zero, such that the matrix pencil A(λ) =∑r
i=0 λiAi loses column rank and the corresponding nonzero complex vector x such that A(λ)x = 0.
We call the (r+1)-tuple λ = (λ0, . . . , λr ) an eigenvalue and the corresponding vector x 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, H2 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 H2 optimal
model reduction problem, to 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 m2 × n2
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 m(m−1)
2 × 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 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 of dimension n(n+1)
2 × n(n+1)
2 . A general solution algorithm is presented and
numerical examples are given to outline the procedure of the proposed algorithm.