Let $A,E \in \C^{n \times n}$ be two given matrices, where rank$E=r < n$. Matrix $E$ is written in the form $E=UV^H$ where $U,V \in \C^{n \times r}$ have rank $r$. ~$0$ is an eigenvalue of $E$ with algebraic (resp. ~geometric) multiplicity $m$ ($g =n-r \leq m$). We consider the pencil $P_z=(A-zI)+t E$, defined for $t \in \C$ and depending on the complex parameter $z \in \C$. We analyze how its structure evolves as the parameter $z $ varies, by means of conceptual tools borrowed from Homotopic Deviation theory. As an example with $z=0$, the structure of the pencil $A+tE$ is determined by Homotopic Deviation.
