Some Contributions of Homotopic Deviation to the Theory of Matrix Pencils

Let A,E ∈ Cn×n be two given matrices, where rankE = r ≤ n. The matrix E is written in the form (derived from SVD) E = UV H where U, V ∈ Cn×r 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 Pz(t) = (A−zI)+ tE, defined for t ∈ Ĉ = C∪{∞} which depends on the complex parameter z ∈ C. 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 Ĉ, 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 C as |t| → ∞.