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Let $f:\mathbb{C}^6\to\mathbb{C}^6$ be a linear map with characteristic polynomial $ch_f(X)=(X+2)^4(X-1)^2$ and $rank(f+2id)>rank(f+2id)^2=1$, as well as $rank(f-id)=5$.

Assignment: Find the minimal polynomial and all possible Jordan canonical forms of $f$ (Hint: There are $2$ different Jordan canonical forms).

I can see that we have exactly one Jordan Block for the eigenvalue $1$ but, given the above information, it seems to me that the only other things we can conclude are that we have at most $3$ Jordan blocks for the eigenvalue $-2$, with at least one Jordan Block of size $\geq2$ (by elementary canonical form theory) and that gives us a total of $4$ possible Jordan canonical forms, as opposed to the hint above.

Thank you very much in advance.

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