Searching for a counter example for this statement.

Question

Let $A\in \mathbb{C}^{n\times n}$ be similar to $A^2$, then $A=A^2$.

This was in a multiple choice question, I'm trying to construct a counter example, but it's getting complicated for me since I'm trying to use upper triangular or diagonal matrices, I'm trying to fill in the diagonal with different numbers so I can get different eigen values which makes my life easier to decide similarity between $A$ and $A^2$. but It's not really working for me.
I would appreciate any help, and it would mean alot to explain to me your steps of constructing the counter example.
Thanks in advance.

Answer 1

Maybe take, $A=\begin{pmatrix}1 &0\\1&1\end{pmatrix}$, $A^2=\begin{pmatrix}1 &0\\2&1\end{pmatrix}$. The matrices are similar but not equal.
I am not sure if you can use diagonal matrices as their squares will not give you similarity.

Answer 2

First notice, that the Eigenvalues of $A^2$ are the squares of the Eigenvalues of $A$. Also, since $A$ and $A^2$ are similar, they have the same Eigenvalues. Or, to rephrase, if we take $\Lambda \subseteq \mathbb{C}$ to be the set of Eigenvalues of $A$, we have $\{\lambda^2 | \lambda \in \Lambda\} = \Lambda$.

Now, try to find a finite set $\Lambda$ with this property and containing at least two values: First, since the set is finite, all elements need to have norm $1$. Visualizing these as points on the unit circle, you can find $\Lambda = \{ e^{\frac{2\pi i}3}, e^{\frac{4\pi i}3} \}$.

Thus you get $A = \begin{pmatrix} e^{\frac{2\pi i}3} & 0 \\ 0 & e^{\frac{4\pi i}3} \end{pmatrix}$ as a counterexample.