Parameter analysis of the structure of matrix pencils by homotopic deviation theory

Let A, E ∈ ℂ^{n ×n} be two given matrices, where rank E = r < n. Matrix E is written in the form E = UV^H where U, V ∈ ℂ^{n ×r} have rank r. 0 is an eigenvalue of E with algebraic (resp. geometric) multiplicity m (g = n – r ≤ m). We consider the pencil P_z = (A – zI) + tE, defined for t ∈ ℂ and depending on the complex parameter z ∈ ℂ. We analyze how its structure [8] evolves as the parameter z varies, by means of conceptual tools borrowed from Homotopic Deviation theory [1, 3]. As an example with z = 0, the structure of the pencil A + tE is determined by Homotopic Deviation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)