Diagonal matrices

Question

Let $D_{n}$ be the set of all n*n complex diagonal matrices. Does there exist a unitary matrix $U$ in $M_{n}(\mathbb{C})$ but not in $D_{n}$ such that $UD_{n}U^{*}= D_{n}$?

Answer 1

Any permutation matrix will work. For instance $$U=\pmatrix{0&1&0&\cdots&0\\ 1&0&0&\cdots&0\\0&0&1&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\cdots&1}.$$