Where is the nearest non-regular pencil?

Abstract This is a first step toward the goal of finding a way to calculate a smallest norm deregularizing perturbation of a given square matrix pencil. Minimal de-regularizing perturbations have geometric characterizations that include a variable projection linear least squares problem and a minimax characterization reminiscent of the Courant-Fischer theorem. The characterizations lead to new, computationally attractive upper and lower bounds. We give a brief survey and illustrate strengths and weaknesses of several upper and lower bounds some of which are well-known and some of which are new. The ultimate goal remains elusive.

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