Matrices that are similar to their squares Suppose that $A$ is an invertible complex $3 \times 3$ matrix such that $A$ and $A^{2}$ are conjugate. What are the possible Jordan forms of $A$?
Again, I have compared eigenvalues for $A$ and $A^{2}$ and resort to cases: if eigenvalues are all distinct, if they are all the same, etc. I might have shared a similar answer previously, but I was wondering if anyone has a better approach to tackling this. 

Hint: You might note that the square of a Jordan block of size $\ge 2$ for eigenvalue $0$ has null space of dimension $2$, so $A$ can't have such a block
in its Jordan form.  On the other hand, the square of a Jordan block for a nonzero eigenvalue $\lambda$ is similar to a Jordan block of the same size for $\lambda^2$.  
