Solving overdetermined eigenvalue problems. We propose a new interpretation of the generalized overdetermined eigenvalue problem (𝐀-λ𝐁)π―β‰ˆ0 for two mΓ—n(m>n) matrices 𝐀 and 𝐁, its stability analysis, and an efficient algorithm for solving it. Usually, the matrix pencil {𝐀-λ𝐁} does not have any rank deficient member. Therefore we aim to compute Ξ» for which 𝐀-λ𝐁 is as close as possible to rank deficient; i.e., we search for Ξ» that locally minimize the smallest singular value over the matrix pencil {𝐀-λ𝐁}. Practically, the proposed algorithm requires π’ͺ(mn 2 ) operations for computing all the eigenpairs. We also describe a method to compute practical starting eigenpairs. The effectiveness of the new approach is demonstrated with numerical experiments. A MATLAB-based implementation of the proposed algorithm can be found at http://www.mat.univie.ac.at/Β neum/software/oeig/.