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



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> 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?
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On the eigen vectors of a diagonalizable matrix
I [mentioned](https://mathoverflow.net/questions/420993/on-the-eigen-vectors-of-a-diagonalizable-matrix#comment1081449_420993) the silly example $U = A = D$ offhand, but, in fact, it's essentially the only example. In general, note that the columns of $U^{-1}$ are the eigenvectors of $A$. So we're asking for, at least, a unitary matrix $U$ that admits the columns of $U^{-1}$ as eigenvectors. But $U U^{-1} = I$, so this means that $U$ must be diagonal.


