Consider I have an arbitrary $n \times n$ Hermitian matrix $A$. I want to derive a "suitable" Toeplitz matrix from $A$. I understand that there may be several ways to get a Toeplitz matrix from $A$ so question may be not clear under this broad term. Is this a problem well defined and there are papers related to it?
I am interested to find the Toeplitz matrix with minimal change in the eigenvectors. Specifically I want find the Toeplitz matrix such that its $L_2$ norm between the eigenvectors of the Toeplitz matrix and eigenvectors of the matrix $A$ is minimal. Can anyone help me how to go about it?
This isn't a full answer to your question, but all toeplitz matrices can be decomposed into the product of a Vandermonde matrix and a diagonal matrix. In this way, your question is about finding the Vandermonde matrix with the smallest principal angle between subspaces to to the eigenvector matrix in question