$T: \mathbb{C}_{2020}[x]\to \mathbb{C}_{2020}[x]$, $\sum_{i=0}^{2020}a_ix^i \mapsto \sum_{i=0}^{2020}a_i(x-1)^i$

$T: \mathbb{C}_{2020}[x]\to \mathbb{C}_{2020}[x]$

$\sum_{i=0}^{2020}a_ix^i \mapsto \sum_{i=0}^{2020}a_i(x-1)^i$

I have to find the presentation matrix of $T$ in order to find Jordan form.

I tried to find it by the standard basis of $\mathbb{C}_{2020}[x]$ but I am little confused about the $\sum_{i=0}^{2020}a_i(x_i-1)^i.$

I cant present the polynomials with the basis standard.

Help please?

Here is how to put it into Jordan form without calculation.

Let $\lambda$ be an eigenvalue of $T$ and $f \in \Bbb C_n[x]$ be an eigenvector for $\lambda$, where $n = 2020$. That is, $T(f) = \lambda f$, i.e. $f(x - 1) = \lambda f$.

By comparing the leading coefficient, we see that $\lambda$ must be equal to $1$. Moreover, $f(x) = f(x - 1)$ can only happen when $f$ is a constant polynomial, by e.g. arguing with the sum of all roots of $f$.

Thus $T$ has only one eigenvalue $\lambda = 1$ and the space of eigenvectors for $\lambda = 1$ is one dimensional. Therefore the Jordan form has only one block, with diagonal and subdiagonal entries equal to $1$ and other entries equal to $0$.