Suppose we have a square $ntimes n$ real matrix $A$ of full rank such that the squares of the elements in each row sum to 1, an $ntimes 1$ vector of variables $x$, and an $ntimes 1$ real vector $a$, such that $Acdot x = a$. 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 $ntimes 1$ 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 $epsilon$?
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