Abstract
Let 
be two given matrices, where rankE = r ≤ n. The matrix E is written in the form (derived from SVD) E = UV
H where 
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 
which depends on the complex parameter 
. 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 
, 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 
as |t| → ∞.
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Chatelin, F., Ahmadnasab, M. (2009). Some Contributions of Homotopic Deviation to the Theory of Matrix Pencils. In: Margenov, S., Vulkov, L.G., Waśniewski, J. (eds) Numerical Analysis and Its Applications. NAA 2008. Lecture Notes in Computer Science, vol 5434. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00464-3_2
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DOI: https://doi.org/10.1007/978-3-642-00464-3_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-00463-6
Online ISBN: 978-3-642-00464-3
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