Some Contributions of Homotopic Deviation to the Theory of Matrix Pencils
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| → ∞.
KeywordsHomotopic Deviation observation point frontier point communication matrix induction matrix matrix pencil Weierstrass Cauchy
Unable to display preview. Download preview PDF.
- 1.Ahmadnasab, M.: Homotopic Deviation theory: A qualitative study. PhD thesis, Université Toulouse 1 and CERFACS, Toulouse, France, October 24 (2007)Google Scholar
- 2.Ahmadnasab, M.: An order reduction method for computing the finite eigenvalues of regular matrix pencils. Technical Report TR/PA/08/23, CERFACS, Toulouse, France (2008)Google Scholar
- 3.Ahmadnasab, M., Chaitin-Chatelin, F.: Parameter analysis of the structure of matrix pencils by Homotopic Deviation theory. In: Proceedings ICIAM 2007. Wiley, Chichester (2007) (to appear)Google Scholar
- 4.Ahmadnasab, M., Chaitin-Chatelin, F.: Matrix pencils under Homotopic Deviation theory. Technical Report TR/PA/07/108, CERFACS, Toulouse, France (2007)Google Scholar
- 6.Chaitin-Chatelin, F.: Computing beyond classical logic: SVD computation in nonassociative Dickson algebras. In: Calude, C. (ed.) Randomness and Complexity, pp. 13–23. World Scientific, SingaporeGoogle Scholar
- 8.Chatelin, F.: Homotopic Deviation in Linear algebra. In: Qualitative Computing: a computational journey into nonlinearity, vol. 7. World Scientific, Singapore (to appear, 2009)Google Scholar