Boutry, Gregory; Elad, Michael; Golub, Gene H.; Milanfar, Peyman The generalized eigenvalue problem for nonsquare pencils using a minimal perturbation approach. (English) Zbl 1100.65035 SIAM J. Matrix Anal. Appl. 27, No. 2, 582-601 (2005). Authors’ summary: This work focuses on nonsquare matrix pencils \(A-\lambda B\), where \(A,B\in{\mathcal M}^{m\times n}\) and \(m>n\). Traditional methods for solving such nonsquare generalized eigenvalue problems \((A-\lambda B)\underline v=\underline 0\) are expected to lead to no solutions in most cases. 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 when \(n=1\) (vector pencils); and (ii) more generally, the \(n>1\) case with the existence of one eigenpair. For both, this paper proposes insight into the characteristic structure of these problems along with practical numerical algorithms toward their solution. We also present a simplifying factorization for such nonsquare pencils, and some relations to the notion of pseudospectra. Reviewer: E. Kreyszig (Ottawa) Cited in 3 ReviewsCited in 19 Documents MSC: 65F15 Numerical computation of eigenvalues and eigenvectors of matrices 15A18 Eigenvalues, singular values, and eigenvectors 15A22 Matrix pencils Keywords:generalized eigenvalue; pseudospectra; nonsquare matrix pencils; minimal perturbation; numerical algorithms; factorization Software:ARPACK; eigs; JDQR; Eigtool; StratiGraph; JDQZ PDFBibTeX XMLCite \textit{G. Boutry} et al., SIAM J. Matrix Anal. Appl. 27, No. 2, 582--601 (2005; Zbl 1100.65035) Full Text: DOI