Um burro
Reputation
Top tag
Next privilege 5 Rep.
Participate in meta
 Dec 15 comment Question regarding the diagonalizability of certain matrices I noticed what you said is basically the same. The fact that $J_A^m=I_n$ implies $J_A$ diagonal follows from this identity easily: en.wikipedia.org/wiki/Jordan_Normal_Form#Powers Thanks. Dec 15 comment Question regarding the diagonalizability of certain matrices I got another approach for the first part: Let $J_A$ be 'the' Jordan Normal Form of $A$. We have $P^{-1}AP=J_A$, for some $P$. It follows that $(J_A)^m=I_n$. From the last equality it will follow that $J_A$ is a diagonal matrix. Dec 15 comment Question regarding the diagonalizability of certain matrices Oh, nevermind, $\lambda$ can't be zero. Let me think about your idea. Dec 15 comment Question regarding the diagonalizability of certain matrices I don't think that's true. Consider $\lambda=0$, its Jordan block is nilpotent, therefore it is diagonal and it doesn't have to be a $1\times 1$ block. EDIT: I wanna thank you for your time before you get fed up with this problem. Dec 15 comment Question regarding the diagonalizability of certain matrices I meant $r((M-\lambda I)^k)=r((M-\lambda I)^{k+1})$. Sorry again. Dec 15 comment Question regarding the diagonalizability of certain matrices Yes, it is the rank. Sorry. Dec 15 comment Question regarding the diagonalizability of certain matrices I don't know that. In my notes the size of the largest block is the smallest natural $k$ such that $r((M-\lambda I)^k)=r((M-\lambda I)^{r+1})$, hence my comment above. Dec 15 awarded Editor Dec 15 revised Question regarding the diagonalizability of certain matrices added 14 characters in body Dec 15 comment Question regarding the diagonalizability of certain matrices The proof of (1) is ridiculously long and it's not on my notes, so I'm guessing I'm supposed to prove it in another way. My idea is to show that the matrices' Jordan Normal Form is a diagonal matrix. To do that it suffices to prove that $r(M-\lambda I)=r((M-\lambda I)^2)$ for every eigenvalue $\lambda$. Dec 15 asked Question regarding the diagonalizability of certain matrices