I am faced with the following problem:

Given endomorphism $f$ whose characteristic polynomial is $$P_c(x) = (x+1)^{10} (x-1)^{10} x^{10}$$ and whose minimal polynomial is $$P_m (x) = (x+1)^5 (x-1)^5 x^5$$ find all the possible Jordan canonical forms knowing that $$\dim(\mbox{ker}(f+id)) = 3, \qquad \dim(\mbox{ker}(f-id))=3, \qquad \dim(\mbox{ker}(f)) = 7$$

I suspect there is no possible Jordan Canonical form since the degree of $x$ in the minimal polinomial is $5$ (and therefore there is a block of dimension 5 for $0$ in Jordan matrix) but there is no way to fill the rest since $10-7+1 = 4 < 5$. Is this correct?

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    $\begingroup$ $10-7=3$ but yes, it seems you are right. $\endgroup$ – Berci May 15 at 12:08
  • $\begingroup$ @Berci Yes, It was a careless mistake, I meant there are 4 numbers between 7 and 10. Thank you! $\endgroup$ – Zanzag May 15 at 12:13

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