Solving Overdetermined Eigenvalue Problems

We propose a new interpretation of the generalized overdetermined eigenvalue problem (A− λB)v ≈ 0 for two m × 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 λ for which A − λB is as close as possible to rank deficient; i.e., we search for λ that locally minimize the smallest singular value over the matrix pencil {A − λB}. The proposed algorithm requires O(mn2) 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/~neum/software/oeig/