If $A^2$ and $B^2$ are similar matrices, do $A$ and $B$ have to be similar?
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.
Solution 1:
No: consider $$ A=\begin{bmatrix}1&0\\0&1\end{bmatrix},\qquad B=\begin{bmatrix}1&0\\0&-1\end{bmatrix} $$
Solution 2:
A little more dramatic, should you want another example:
$A=\begin{bmatrix}0&1\\0&0\end{bmatrix},\qquad B=\begin{bmatrix}0&0\\0&0\end{bmatrix}$