The Generalized Eigenvalue Problem for Nonsquare Pencils Using a Minimal Perturbation Approach

This work focuses on non-square matrix pencils A − λB where A,B ∈ Mm×n and m > n. Traditional methods for solving such nonsquare generalized eigenvalue problems (A−λB)v = 0 are expected to lead to no solutions in most cases. In this paper we propose a different treatment; We search for the minimal perturbation to the pair (A,B) such that these solutions are indeed possible. Two cases are considered and analyzed: (i) The case where n = 1 (vector pencils); and (ii) more generally the n > 1 case with the existence of one eigenpair. For both, this paper proposes insight into the characteristics of the described problems along with practical numerical algorithms towards their solution. We also present a simplifying factorization for such nonsquare pencils, and some relations to the notion of pseudospectra. ∗The Computer Science Department, The Technion Israel Inst. of Techology, Haifa, 32000 Israel †Department of Electrical Engineering, University of California, Santa Cruz CA. 95064 USA