مراد احمدنسب

‎Let $A$ $\in$ $\mathbb{C}^{n \times n} $ be a given matrix, $z\in$ $\mathbb{C} \backslash \sigma(A)$, where $\sigma(A)$ denotes the spectrum of $A$. We consider the problem (P): find $\triangle A \in \mathbb{C}^{n \times n} $ such that $A+\triangle A-zI$ is singular.\\‎ ‎That is:~ find $\triangle A$ such that $z$ is an eigenvalue of $A+\triangle A$.\\‎ ‎With no further assumption on $\triangle A$, the problem (P) has an infinity of solutions. This is the framework of the well-known normwise backward analysis which looks for $\triangle A$ with minimum norm. \\‎ ‎In homotopic deviation theory, $\triangle A$ has a prescribed structure $E$ such that $\triangle A=tE$, $t\in \mathbb{C}$, and the deviation matrix $E\in \mathbb{C}^{n \times n}$ is fixed.The value $z$ is the eigenvalue of at most $n$ matrices $A+t_iE$, $i=1,..,r \le n$, with $t_i \in \mathbb{\hat C}=\mathbb{C} \cup {\infty}$. Modifications $\triangle A$ of A with a prescribed structure seem to play an important role in our current understanding of the evolution of living organisms.\\‎ ‎The detailed comparison of the two backward analyses, normwise and homotopic, shows that the latter is computationally much richer than the first.‎ ‎The deviation matrix $E$ of rank $r \le n$ can be written, using the Singular Value Decomposition, as $E=UV^{H}$, both $U, V$ $\in$ $\mathbb{C}^{n \times r}$ of rank $r$.~~Set $M_{z}=V^{H}(zI-A)^{-1}U$ for any $z$ in $\mathbb{C}\backslash$ $\sigma(A)$:‎ ‎~~z is an eigenvalue of $A+tE$ iff $t\mu_{z}=1$,for $\mu_{z}$ $\in$ $ \sigma(M_{z})$.~ $R(t,z)=(A+tE-zI)^{-1}$ is analytic in $t$ (resp. $1/t$) for $|t|<1/\rho(M_{z})$ (resp. $|t|>\rho(M_{z}^{-1}$)) for z such that $M_{z}$ has rank $r$.‎ ‎~~There are important consequences when $E$ is {\em singular}.~When $r