Given $A,B\in\mathbb R^{n\times n}$ such that there is an $U\in\mathbb R^{n\times n}$ with $UU'=I$ and $UAU'=B$.
Suppose each entry of $V$ is within $\pm\epsilon$ of each entry of $U$ is there a way to bound $$\|VAV'-B\|_2^2?$$
Suppose we can create a matrix $\widehat B$ such that each entry of $B$ can be modified (added or multiplied) by some polynomial of $\epsilon$ (we can have different polynomial for each entry of $B$) then for what choices of functions can we get minimum $$\|VAV'-\widehat B\|_2^2?$$