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چکیده
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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
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