  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.
