I'm working on the following problem:
Let $V_{100}$ denote the $\mathbb{C}$-vector space consisting of all polynomials in $\mathbb{C}[x]$ of degree $\leq 100$. Let $T$ be the linear operator on $V_{100}$ defined by $T(f(x)) = 2f(x) - f'(x)$. Find the Jordan canonical form of $T$.
In order to find the Jordan canonical form of $T$, I know that I need the minimal polynomial of $T$ and the characteristic polynomial of $T$. To this point, I tried to find an annihilating polynomial of $T$ first. But, the annihilating polynomial for $T$ that I found is $f(x) = x^{101} - 2^{101}x$. This seems to be too complicated of an annihilating polynomial to deduce what the minimal and characteristic polynomials have to be. Certainly, it doesn't seem to be a nilpotent operator, which is an easier case.
Is there another approach that can tell me what the minimal and characteristic polynomials should be ? I feel as if my approach (sort of brute force) is giving me lots of trouble.
Thanks!
Since $Af:=(T-2\operatorname{Id})f=-f'$, we have $A^{101}=0\ne A^{100}$. Therefore the minimal polynomial of $A$ is $x^{101}$, the minimal polynomial of $T$ is $(x-2)^{101}$ and the Jordan form of $T$ is the $101\times101$ Jordan block for the eigenvalue $2$.