Let $A,E \in \bbbc^{n \times n}$ be two given matrices, where rank$E=r \leq n$. The matrix $E$ is written in the form (derived from SVD) $E=UV^H$ where $U,V \in \bbbc^{n \times r}$ have rank $r \leq n$. For $0 < r < n$, ~$0$ is an eigenvalue of $E$ with algebraic (resp. ~geometric) multiplicity $m$ ($g =n-r \leq m$). We consider the pencil $P_z(t)=(A-zI)+t E$, defined for $t \in \hat{\bbbc}=\bbbc \cup \{ \infty \}$ which depends on the complex parameter $z \in \bbbc$. We analyze how its structure evolves as the parameter $z $ varies, by means of conceptual tools borrowed from Homotopic Deviation theory \cite{mythesis,QC_book_2009}. The new feature is that, because $t$ varies in $\hat{\bbbc}$, we can look at what happens in the limit when $|t| \to \infty$. This enables us to propose a remarkable connection between the algebraic theory of Weierstrass and the Cauchy analytic theory in $\bbbc$ as $|t| \to \infty$.
