Suppose that $e_1, \cdots, e_n$ are the standard vectors of the Euclidean space   $\mathbb{R}^n$. 

Let us consider the backward shift operator $T:\mathbb{R}^n\to \mathbb{R}^n$ given by $Te_k=e_{k-1}$ if $k\geq2$ and $Te_1=e_n$.   

Let $A:\mathbb{R}^n\to \mathbb{R}^n$ be the operator given by $Ae_3=Ae_4=e_1$ and $Ae_k=0$ for any  other $k$.  

It is well-known that the coulmns of the discrete Fourier matrix are just the eigenvectors of $T$.  What about the (analytical proof) eigenvectors of $T+A$?