Do $\sqrt{A}BA^*B\sqrt{A}$ and $ABA^*B$ have the same eigenvalues? where $A$ is a 4 by 4 positive-semidefinite hermitian matrix and $\text{tr}A=1$ and $B=\begin{bmatrix} 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{bmatrix}$.
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1See here https://math.stackexchange.com/questions/311342/do-ab-and-ba-have-same-minimal-and-characteristic-polynomials. Cyclic permutation of a product of square matrices always preserves its eigenvalues, including their multiplicities. – Badam Baplan Nov 29 '18 at 06:22