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The generalized eigenvalue problem for nonsquare pencils using a minimal perturbation approach. (English) Zbl 1100.65035

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.

MSC:

65F15 Numerical computation of eigenvalues and eigenvectors of matrices
15A18 Eigenvalues, singular values, and eigenvectors
15A22 Matrix pencils
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