Finding an operator that satisfies a given Minimal and characteristic Polynomial?

Question

I am studying for an upcoming Linear Algebra exam. I am going through the questions from an old exam the instructor gave out, and I have come to this problem:

Give an example of an operator on a complex vector space with characteristic polynomial $(z-2)^3 (z-3)^3$ and with minimal polynomial $(z-2)^3(z-3)^2$.

Now I know that the matrix for this operator must have three $2$'s and three $3$'s down the diagonal, and I know the minimal polynomial divides the characteristic, but I don't know much else. This is in the same chapter as Jordan form, so I think a solution might have to do with Jordan blocks, but I don't have enough intuition about those to get it.

Any help here? :)

Answer 1

$$ \left( \begin{array}{cccccc} 2 & 1 & 0 & 0 & 0 & 0 \\ 0 & 2 & 1 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 3 & 1 \\ 0 & 0 & 0 & 0 & 0 & 3 \end{array} \right). $$

Answer 2

Take a matrix with three Jordan blocks $\pmatrix{A & 0 & 0\cr 0 & B & 0\cr 0 & 0 & C\cr}$. Note that its characteristic polynomial is the product of the characteristic polynomials of $A$, $B$ and $C$. Choose $A$ to have characteristic and minimal polynomial $(z-2)^3$. Choose $B$ to have characteristic and minimal polynomial $(z-3)^2$. What do you suppose $C$ should have?