<s>[INST] How to prove that A and B are similar Let be $$A=\begin{pmatrix}
\frac{-3}{2} & 2 & \frac{-1}{2} \\
\frac{-1}{2} & 0 & \frac{1}{2} \\
\frac{1}{2} & -2 & \frac{3}{2}
\end{pmatrix},
B=\begin{pmatrix}
0 & 1 & 0 \\
0 & 0 & 1 \\
0 & 0 & 0
\end{pmatrix}$$
Prove that A and B are similar.
I know if we can find a matrix $P$ so that $A=P^{-1}BP$ we have that they are similar, but I haven't found anywhere how to find such $P$. Can someone give me a hint?
 [/INST] The easiest way I know would be to show that $A^2$ and $B^2$ are nonzero, but $A^3=B^3=0$.  There is only one $3$-by-$3$ nilpotent matrix of nilpotency index $3$, up to conjugation.
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