Suppose we have a square $n\times n$ real matrix $A$ of full rank such that the squares of the elements in each row sum to 1, an $n\times 1$ vector of variables $x$, and an $n\times 1$ real vector $a$, such that $A\cdot x = a$. We can of course take the inverse of $A$ to solve uniquely for $x$. 


My question is as follows: suppose we do not know $a$ exactly, but only up to additive error epsilon: that is, we know $a'$ such that $a' = a + error$, where $error$ is a real $n\times 1$ vector with each component in the range $[-\epsilon,\epsilon]$. Vector $x$ is no longer uniquely determined. However, we can solve for some $x'$ such that $x' = x + error'$. My question is, what can we say about the magnitude of the components of $error'$, and how they relate to $\epsilon$?



Solving a noisy set of linear equations.
To be a little more precise, the assumption here is that $\| error \|\_\infty \le \epsilon$ and it seems you want to bound $\| error' \|\_\infty$. So from $error' = A^{-1} error$ it follows that $\| error'\|\_\infty \le \| A^{-1} \|\_\infty \epsilon$. Here $\| A^{-1} \|\_\infty$ is the operator norm of $A^{-1}$ induced by the $\infty$-norm on $\mathbb{R}^n$, which is easily expressed in terms of the entries of $A^{-1}=[b\_{ij}]$:
$$\| A^{-1} \|\_\infty = \max\_{1\le i\le n} \sum\_{j=1}^n |b\_{ij}|.$$


