unitary decomposition of diagonal matrices

Question

Let $D \in \mathbb{R^{n,n}}$ be a diagonal Matrix with diagonal entries $d_{ii} \neq 0$. Let $\tilde{D} \in \mathbb{R}^{n,n}$ be a diagonal matrix with the same diagonal entries that could be transposed in any way (for example $\tilde{d}_{11}=d_{22}$). Does there exist a unitary matrix $U$ such that $$UDU^H=\tilde{D}$$?

Answer 1

I am not sure if I understand your condition. It seems that your are looking for something like $$ \begin{pmatrix} 0&1\\1&0 \end{pmatrix} \begin{pmatrix} a&0\\0&b \end{pmatrix} \begin{pmatrix} 0&1\\1&0 \end{pmatrix}^{-1} = \begin{pmatrix} b&0\\0&a \end{pmatrix}, $$ where $\begin{pmatrix} 0&1\\1&0 \end{pmatrix}$ is unitary. For general $n\times n$ matrices we can use matrix blocks to construct such examples.