TITLE: If $A^2$ and $B^2$ are similar matrices, do $A$ and $B$ have to be similar?
QUESTION [18 upvotes]: I know that the converse is true; that is, if A and B are similar matrices, then $A^2$ and $B^2$ are similar . However, I'm not sure about the reverse.

REPLY [14 votes]: A little more dramatic, should you want another example:
$A=\begin{bmatrix}0&1\\0&0\end{bmatrix},\qquad
B=\begin{bmatrix}0&0\\0&0\end{bmatrix}$