# find minimal polynomial of $T(p)=p'+p$

I'm trying to solve the following question:

let $T: \mathbb C_n[x] \to \mathbb C_n[x]$, $T(p)=p'+p$

find the characteristic and minimal polynomial of $T$.

What I'm trying to do is the following: I saw how $T$ interacts with the standard basis of polynomials, $1,x,x^2,...x^n$, and I wrote down $T$ as a matrix so I can easily find its characteristic polynomial.

Lucky for us, the matrix $T$ is triangular:

$T=\begin{pmatrix} 1 & 1 & 0 & 0 & 0&\cdots & 0 \\0 & 1 & 2 & 0 &0&\cdots & 0 \\0&0&1&3&0&\cdots & 0\\ \vdots & \vdots & \vdots &\ddots &\ddots &\cdots & 0\end{pmatrix}$

I'm not that good at mathjax, basically, below the diagonal there are only $0$. on the diagonal only $1$, and at index $(k-1,k)$ there is $k-1$.

It is very easy to see that the characteristic polynomial $p_T(x)=(x-1)^{n+1}$.

The minimal polynomial is harder to find. I know they are equal because I looked at the answers, but how do I prove it?

Note that $T-1$ maps $p\mapsto p'$. Clearly $(T-1)^n\ne0$, as is witnessed by $x^n\mapsto\ldots\mapsto n!$.