Diagonal Matrix

Question

Suppose that $A$ is a $n \times n$ complex normal matrix. We know that there is a unitary matrix $U$ such that $U^{-1}AU$ is a diagonal matrix. My question is as follows: In the process of diagonalization, is there a unitary matrix $U$ such that $UAU$ is a diagonal matrix?

Answer 1

The Eigendecomposition of $A$ is $A=U\Lambda U^{-1}$ where $\Lambda$ is diagonal and $U$ is unitary. Hence, $U^{-1}=U^*$ and $U^{-1}AU=\Lambda$

In order to $UAU=\Lambda$ hold, one needs to have $U^{-1}=U$, that is $U$ should be an involutory matrix: $U^{2}=I$.

The above conditions can happen jointly when $U=I$. That is, when $A$ is diagonal already, But it is not necessary. For instance if $U=\begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix}$.