# Are they similar matrix

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

• Why negative vote – Vasanth Kris Dec 10 '18 at 6:25
• Welcome to MSE! It is more likely that you'll receive an answer if you showed us that you've made an effort – Vee Hua Zhi Dec 10 '18 at 6:38
• Sorry i dont know how to start – Vasanth Kris Dec 10 '18 at 6:39
• Oh don't mind the downvoters, they have a habit of downvoting posts that doesn't show efforts (and without explaining why) – Vee Hua Zhi Dec 10 '18 at 6:39
• I've made up for it by upvoting. Just make sure you show that you've tried next time – Vee Hua Zhi Dec 10 '18 at 6:39

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?
• 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. – Mark Dec 10 '18 at 6:48