Um burro

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seen Dec 16 '12 at 11:25

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
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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