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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? :)

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    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.2012-11-27

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$ \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). $

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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?

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    Robert, could you please take a look at http://math.stacke$x$change.com/questions/244038/inhomogeneous-equation ? So far I have written to the author, we agree that the part about the degree is not quite right, but I cannot seem to settle on a fi$x$. One possibility is the degree of the characteristic value in the minimal polynomial instead of the characteristic polynomial, but I'm not sure even of that. Maybe that as an upper bound?2012-11-27