Determine if is possible to represent the matriz $$\begin{equation} \begin{pmatrix} i & 0 & 2\\ 2i & 0 & -1\\ 0 & -(1+2i) & 0 \end{pmatrix} \end{equation}$$ like $A=\lambda U$ where $U \in \mathbb{C}^{n \times n}$ is an unitary matrix and $\lambda \in \mathbb{C}$. Is possible have $\overline{\lambda}=\lambda$?
I took a eigenvector $v$ of $A$ associated to the eigenvalue $c$, so $Av=cv$. Since se want that $ A= \lambda U$ so $cv= \lambda U v$. Then $ Uv = \frac{c}{\lambda}v$ for $|| \lambda || \neq 0$, therefore $U$ and $A$ have the same eigenvectors.