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Do $\begin{bmatrix} 0&i&0\\0&0&1\\0&0&0 \end{bmatrix} $ and $\begin{bmatrix} 0&0&0\\-i&0&0\\0&1&0 \end{bmatrix} $ are similar.Is this True/false

Clearly both are nilpotent and one is conjucate transpose of other but how to know if they are similar.i'm stuck. Please help me

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  • $\begingroup$ Why negative vote $\endgroup$ – Vasanth Kris Dec 10 '18 at 6:25
  • $\begingroup$ Welcome to MSE! It is more likely that you'll receive an answer if you showed us that you've made an effort $\endgroup$ – Vee Hua Zhi Dec 10 '18 at 6:38
  • $\begingroup$ Sorry i dont know how to start $\endgroup$ – Vasanth Kris Dec 10 '18 at 6:39
  • $\begingroup$ Oh don't mind the downvoters, they have a habit of downvoting posts that doesn't show efforts (and without explaining why) $\endgroup$ – Vee Hua Zhi Dec 10 '18 at 6:39
  • $\begingroup$ I've made up for it by upvoting. Just make sure you show that you've tried next time $\endgroup$ – Vee Hua Zhi Dec 10 '18 at 6:39
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Hint: both matrices have the same characteristic polynomial $p(x)=x^3$, and for both that is also the minimal polynomial. What can you conclude about their Jordan canonical forms?

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  • $\begingroup$ I dont know jordan form any other way? $\endgroup$ – Vasanth Kris Dec 10 '18 at 6:41
  • $\begingroup$ Well, that's one of the reasons why people on this forum ask to show what you tried-because that way it will also be clear how much do you know about the problem. I'll try to think of another solution. $\endgroup$ – Mark Dec 10 '18 at 6:44
  • $\begingroup$ Btw what is answer true or false.... Next time if i ask i will add details as much as i xan $\endgroup$ – Vasanth Kris Dec 10 '18 at 6:46
  • $\begingroup$ They are similar. Matrices of order at most $3\times 3$ (not true for higher dimension matrices) are similar if and only if their characteristic and minimal polynomials are the same. I never thought about proving this lemma without Jordan form though. $\endgroup$ – Mark Dec 10 '18 at 6:48

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