# Canonical Jordan form contradiction

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?

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