Let $X$ and $Y$ be square non-symmetric matrices of the same size. Assume that their eigenvalues are close in the sense that there exists a small $\varepsilon>0$ such that, for any eigenvalue $\lambda$ of $X$, there exists an eigenvalue $\mu$ of $Y$ such that $|\lambda-\mu|\le \varepsilon$.

Now, let $D$ be a diagonal matrix of bounded components. Can we say that eigenvalues of $DX$ and $DY$ are also close in the same sense (with $\varepsilon$ replaced by $C\varepsilon$ for some $C>0$)? References are greatly appreciated.