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

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? :)

• Google "companion matrix". – DonAntonio Nov 27 '12 at 20:55
• So the companion matrix gives a matrix having the given polynomial as its minimal polynomial? How do I account for the fact that the characteristic polynomial has an extra factor of $(z-3)$ then? – Jeff Fuller Nov 27 '12 at 20:59
• Well, then you have to take a matrix of order one more than the companion one, right? From here you're on your own as I'm not sure whether there's an easy path to that. Perhaps there's some trick or method but I don't know/remember it now. – DonAntonio Nov 27 '12 at 21:02

$$\left( \begin{array}{cccccc} 2 & 1 & 0 & 0 & 0 & 0 \\ 0 & 2 & 1 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 3 & 1 \\ 0 & 0 & 0 & 0 & 0 & 3 \end{array} \right).$$
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?
• About $C$, what minimal polynomial should it have? I am a little confused about that actually. – Jeff Fuller Nov 27 '12 at 21:17
• Oh but wait. C should just have $z-3$ as its minimal polynomial, right? Aka C should be just the single value 3 on the diagonal. – Jeff Fuller Nov 27 '12 at 21:19