Some Contributions of Homotopic Deviation to the Theory of Matrix Pencils

  • Françoise Chatelin
  • Morad Ahmadnasab
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5434)


Let Open image in new window be two given matrices, where rankE = r ≤ n. The matrix E is written in the form (derived from SVD) E = UV H where Open image in new window have rank r ≤ n. For 0 < r < n,  0 is an eigenvalue of E with algebraic (resp.  geometric) multiplicity m (g = n − r ≤ m).

We consider the pencil P z (t) = (A − zI) + t E, defined for Open image in new window which depends on the complex parameter Open image in new window . We analyze how its structure evolves as the parameter z varies, by means of conceptual tools borrowed from Homotopic Deviation theory [1,8]. The new feature is that, because t varies in Open image in new window , we can look at what happens in the limit when |t| → ∞. This enables us to propose a remarkable connection between the algebraic theory of Weierstrass and the Cauchy analytic theory in Open image in new window as |t| → ∞.


Homotopic Deviation observation point frontier point communication matrix induction matrix matrix pencil Weierstrass Cauchy 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Françoise Chatelin
    • 1
  • Morad Ahmadnasab
    • 2
    • 3
  1. 1.Université Toulouse 1 and CERFACSToulouse Cedex 1France
  2. 2.Department of MathematicsUniversity of Kurdistan, Pasdaran boulevardSanandajIran
  3. 3.CERFACSToulouse Cedex 1France

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