For an arbitrary $NXN$ Hermitian matrix $A$. I want to derive a Toeplitz matrix from $A$ such that eigenvectors of both matrix has minimal change. Specifically I want find the Toeplitz matrix such that its L2 norm between the eigenvectors of the Toeplitz matrix and eigenvectors of the matrix $A$ is minimum. Is there any other alternative method than numerically search for the matrix; such search would be very long?

I am aware of some work done related to perturbation 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 $A$?