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Let $\mathbf{U}_1$ and $\mathbf{U}_2$ be two arbitrary unitary matrices and $\mathbf{D}$ be a diagonal matrix. What is the solution of the following optimization problem? \begin{align} \min_{\mathbf{D}}\|\mathbf{U}_1\mathbf{D}\mathbf{U}_2-\mathbf{I}\|_F , \end{align} where $\mathbf{I}$ is the identity matrix and $\|\cdot\|_F$ is the Frobenius norm.

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    $\begingroup$ Since $U_1$ is unitary and the Frobenius norm in unitarily equivalent, the objective is equal to $\|DU_2 - U_1^H||_F$ and thus, decouples over the rows of $U_2$. $\endgroup$
    – Dirk
    Commented Feb 7, 2022 at 9:39
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    $\begingroup$ Do that again to get to $D-U_1^H U_2^H $ and you've got an explicit expression for the elements of $D$ ... $\endgroup$
    – Michael Engelhardt
    Commented Feb 7, 2022 at 15:06

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