Solving Overdetermined Eigenvalue Problems

Abstract

We propose a new interpretation of the generalized overdetermined eigenvalue problem $\left({\bf A} - \lambda {\bf B}\right){\bf v} \approx 0$ for two $m\times n$ ($m > n$) matrices ${\bf A}$ and ${\bf B}$, its stability analysis, and an efficient algorithm for solving it. Usually, the matrix pencil $\{{\bf A} - \lambda {\bf B}\}$ does not have any rank deficient member. Therefore we aim to compute $\lambda$ for which ${\bf A} - \lambda {\bf B}$ is as close as possible to rank deficient; i.e., we search for $\lambda$ that locally minimize the smallest singular value over the matrix pencil $\{{\bf A} - \lambda {\bf B}\}$. Practically, the proposed algorithm requires ${\cal O}(mn^2)$ 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/ neum/software/oeig/.