I'm trying to understand the relation (if any) between the eigenvectors of similar matrices and in particular of a matrix and its diagonalization.
Given $A,D\in M^F_{n\times n}$ and invertible $P$ such that $P^{-1}AP=D$ then $AP=PD$ and the eigenvectors of A are the columns of $P$ because $AP_i=\lambda_iP_i$ and $P$ is a change of basis matrix from whatever basis $A$ is in to whatever basis $PD$ is in. $D$ itself is obviously diagonalizable, and its eigenvectors are the columns of $I$ which won't equal $P$ unless $A=D$. And that's as far as I can get at the moment.
EDIT
As Marvis noted, the heart of the question is what is the relationship ( if any ) between the eigenvectors of two similar matrices.