Jordan Canonical Form set of polynomials over complex field

Question

N: $P_4(C) \to P_4(C)$ defined by $N(p) = p''- 3p'$

Find the canonical form and a canonical basis for the mapping of $N$.

I am not sure how to compute this when it is over $\mathbb{C}$ instead of over $\mathbb{R}$. I tried looking on the internet for set of polynomials over a field but could not find anything helpful.

A full solution would be lovely to show how you would even compute $N^2$, $N^3$, etc.

Answer 1

I assume that $P_4(\mathbb{C})$ is the vector space of the polynomials of the complex variable $X$ of degree $\leq 4$ and $p',p'',\cdots$ are the derivatives of the polynomial $p$.

The matrix of $N$ is $5\times 5$ and is nilpotent. Indeed, $N^5=0$ and $N^4\not= 0$ (because $N^4(X^4)\not= 0$).

Then the jordan form of $N$ is $J_5$, the nilpotent jordan block of dimension $5$. A basis that "Jordanizes" $N$ is $\{N^4(X^4),N^3(X^4),N^2(X^4),N(X^4),X^4\}$.