Computing rank-deficiency of rectangular matrix pencils

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Rectangular matrix pencils arise in many contexts in Linear Control Theory, for example the PBH Test for Controllability and the computation of transmission zeros. In this paper, we examine a method to find the distance from an arbitrary given pencil to a nearby rank-deficient one, in light of the fact that rectangular pencils are generically full-rank. We propose an experimental computational method that exhibits quadratic convergence to a local minimum of this distance function. This partially answers the question of the existence of a rank-deficient pencil in a neighborhood of a given pencil. We use the Algorithm to illustrate some limitations of previous algorithms to measure this distance.

Original languageEnglish (US)
Pages (from-to)207-214
Number of pages8
JournalSystems and Control Letters
Issue number3
StatePublished - Sep 1987

Bibliographical note

Funding Information:
* This research was partially supported by NSF Grants DCR-8420935 and DCR-8519029.


  • Algebraic structure
  • Genericity
  • Matrix pencils
  • Numerical methods
  • Rank-deficiency


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