Does the matrix $U$ that diagonalize another matrix $M$ also diagonalize the derivative of $M$?

Question

I have a $3\times3$ Hermitian matrix $M$. All the elements of $M$ are a function of variable $x$. There is a unitary matrix $U$ that diagonalize $M$, i.e. $$ D=U^\dagger MU $$ I wonder if the same $U$ matrix always diagonalize $\frac{dM}{dx}$?

Answer 1

No. Consider the special case where $D$ is a constant real diagonal matrix, $U=e^{tK}$ for some constant real skew-symmetric matrix $K$ and $t$ is a real parameter. We have \begin{aligned} U^\dagger \dot{M} U &=U^\dagger \dot{M} U\\ &=U^\dagger (\dot{U}DU^\dagger + U\dot{D}U^\dagger + UD\dot{U}^\dagger) U\\ &=U^\dagger (UKDU^\dagger + 0 - UDKU^\dagger) U\\ &=KD-DK, \end{aligned} which is not a diagonal matrix when $K\ne0$ and $D$ has distinct nonzero diagonal elements.