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Considerable literature exists in linear algebra to solve the generalized eigenvalue, eigenvector problem (F - λ G)v = 0 where F, G ∈ ℜ(s × s), are square matrices. However, a number of applications lend themselves to the case where F, G ∈ ℜ(s × t), and s ≠ t. The existing methods cannot be used for such non-square cases. This research explores structural decomposition of a matrix pencil (F - λ G), s ≠ t to compute finite values of λ for which rank(F - λ G) < min(s,t). Moreover, from the decomposition of the matrix pencil, information about the order of λ at infinity, the Kronecker row and column indices of a matrix pencil can also be extracted. Equally important is the computation of non-zero vectors w ∈ ℜ(1 × s) and v ∈ ℜ(t × 1) corresponding to each finite value of λ, such that w(F - λ G) = 0 and (F - λ G)v = 0. Algorithms are developed for the computation of λ, w, and v using numerically efficient techniques. Proposed algorithms are applied to problems encountered in system theory and illustrated by means of numerical examples.