TITLE: Can a perturbation of a matrix product always be represented as product of perturbations of its factor matrices?
QUESTION [6 upvotes]: 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?

REPLY [3 votes]: In numerical analysis lingo, you are more or less asking if matrix multiplication is backward stable. The answer seems to be no: see Section 3.5 of Higham, Accuracy and stability of numerical algorithms. I am unable to locate an explicit counterexample quickly, though.