Let $A$ and $B$ be two diagnolizable square block Toeplitz matrices with the same size but different generating symbols, and their spectra are approximately the same, that is $\sigma(A)=\sigma(B)+O(L^{-1})$ where $L$ is the size of the two matrices. A typical case is that when set $L=100$ only $4$ eigenvalues among all differ slightly, and all other eigenvalues are the same.
I want to know if there is a method to compute a similarity transformation or at least an 'approximate similarity transformation' to connect $A$ and $B$?