I am not sure if this is even a valid question, but it will help to clear my doubts anyway.
For context, check out Violating the Thermodynamic Uncertainty Relation in the Three-Level Maser.
So I am given two (Hermitian) matrices - $$H=\begin{pmatrix} \omega_l & \epsilon e^{i\omega_dt} & 0\\ \epsilon e^{-i\omega_dt} & \omega_u & 0\\ 0 & 0 & \omega_x \end{pmatrix},$$ ('$t$' is some parameter (time)) and, $$X=\begin{pmatrix} \omega_l & 0 & 0\\ 0 & \omega_l+\omega_d & 0\\ 0 & 0 & \omega_x \end{pmatrix}.$$ With this Hermitian matrix $X$ we form a Unitary matrix: $e^{iXt}$.
Now my question is - how to find the new matrix: $H^{rot}=e^{iXt}He^{-iXt}?$
To be exact, I know how to find the matrix $XH$, but tell me how to find the other matrix $e^{iXt}H.$