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?