Given $A=BC$ where $A\in\mathbb{R}^{m\times n}$ and for some $B\in\mathbb{R}^{m\times k}, C\in\mathbb{R}^{k\times n}$. We assume that $k>=\min(m,n)$ so that this decomposition always exists for any matrix $A\in\mathbb{R}^{m\times n}.$


Can we prove that any perturbation $\bar{A}$ of $A$ can be represented as the product of two perturbations of $B$ and $C$ ?


Intuitively i think it should be possible but i cannot prove/disprove it. If not possible then what conditions are needed?



Can a perturbation of a matrix product always be represented as product of perturbations of its factor matrices?
The condition you want is exactly that the matrix multiplication map be locally open at the pair $(B,C)$. This is the topic of the recent paper [Where is matrix multiplication locally open?](http://www.sciencedirect.com/science/article/pii/S0024379516306036) by Behrends. The paper contains a complete characterization in Theorem 2.5. According to that theorem, if we let $s$ and $t$ be the ranks of $B$ and $C$, respectively, and let $t = t\_1 + t\_2$ where $t\_1 = \dim (\operatorname{range} C \cap \ker B)$, then matrix multiplication is open at $(B,C)$ iff $t\_2\leq k-m$ or $n-t\_1\leq k-s$.


