Let us consider the space $M_n(\mathbb{C})$. By a unitary matrix $U=(u_{ij})$ we mean that $U^{-1}=(\overline{u_{ji}})$.

Q. Let $U$ be a unitary matrix. I am looking for the pairs of matrices $(D,A)$ satisfying the following conditions: (1) $D$ is a diagonal matrix in $M_n(\mathbb{C})$, (2) $UAU^{-1}=D$, (3) $U$ and $A$ have the same eigen-vectors. In general case, does there exist such a pair? How can we find a formula to produce all such of the pairs?