For an arbitrary $N\times N$ Hermitian matrix $A$, I want to derive a Toeplitz matrix from $A$ such that the eigenvectors of both matrices have minimal change.

Specifically I want find the Toeplitz matrix such that the $L^2$ norm between the eigenvectors of the Toeplitz matrix and eigenvectors of the matrix $A$ is minimal. Is there any alternative method other than searching numerically for the matrix? What is the computational cost of such such search?

I am aware of some work done related to perturbations of Toeplitz matrices, in addition eigenvectors of banded toeplitz matrix is studied, but the matrix I want in my application is not banded. I would appreciate any suggestion.

Edit: Is the problem tractable/solvable/realistic if we are given a sequence of matrices $A^n$ instead of $A$?