Minimal polynimomial and operator If given a (monic) polynomial $p$ it is possible to find an operator $T$ so that $p$ is its minimal polynomial. I can do it for easy polynomials (like $x^2$) with trying. I was wondering: is there a method to apply to find such operator?
 A: Great question.  The concept you're looking for here is the companion matrix, $C(p)$, of a monic polynomial $p$.
A: As Jared notes, you can take the companion matrix of the polynomial which will in fact have both minimal and characteristic polynomial equal to $p$.
More generally, given polynomials $p$ and $q$ in which $p\mid q$ and every irreducible factor of $q$ appears in $p$, you can find an operator $T$ which has minimal polynomial $p$ and characteristic polynomial $q$. You can do this by considering the Jordan Canonical Form of the operator and selecting the Jordan blocks accordingly.
Edit: In response to OP's comments.
You must have two $0$s appear on the diagonal in the same block for the minimal polynomial to be $x^2$. This means that the size of the largest block is size $2$.
The last diagonal entry also has to be zero since all the eigenvalues of the operator must be zero. All in all, you have something of the form
$$T=\begin{pmatrix}0&1&0\\0&0&0\\0&0&0\end{pmatrix}$$
Note that this operator is uniquely determined (up to similarity).
