مراد احمدنسب

Let $A,E \in \C^{n \times n}$ be two given matrices, where rank$E=r < n$. The matrix $E$ is written in the form (derived from SVD) $E=UV^H$ 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 $t \in \hat{\C}=\C \cup \{ \infty \}$ and depending on the complex parameter $z \in \C$. We analyze how its structure \cite{Gant} evolves as the parameter $z $ varies, by means of conceptual tools borrowed from Homotopic Deviation theory \cite{mythesis,acm:05}.‎ ‎The new feature is that, because $t$ varies in $\hat{\C}$, we can look at what happens in the limit when $|t| \to \infty$. As an example with $z=0$, the structure of the pencil $A+tE$ is determined by Homotopic Deviation when $0 \not \in \sigma(A)$.‎