On Pseudospectra, Critical Points, and Multiple Eigenvalues of Matrix Pencils
SIAM Journal on Matrix Analysis and Applications
Overconstrained linear estimation of radial distortion and multi-view geometry
ECCV'06 Proceedings of the 9th European conference on Computer Vision - Volume Part I
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This work focuses on nonsquare matrix pencils $A - \lambda B$, where $A,B \in {\cal M}^{m \times n}$ 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. 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 when $n=1$ (vector pencils); and (ii) more generally, the $n1$ 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 toward their solution. We also present a simplifying factorization for such nonsquare pencils, and some relations to the notion of pseudospectra.