I need to minimize an expectation value for the Hermitian matrix $M$ and an arbitrary normalized vector $v$:
$$\min(\bar{v} Mv)$$
Here $v$ is fixed and normalized: $\bar{v} v=1$ and $\bar{v}$ is the complex conjugate of $v$ but $M$ has some free parameters inside and I need to optimize them. I thought if I minimize the trace I can minimize the desired equation because by spectral decomposition of $M=U^\dagger M_D U $
$$(\bar{v} Mv)=\bar{v'}M_D v'=\sum_m\bar{v'} mv'= \operatorname{Tr}(M)$$
Here $M_D$ is the diagonal matrix composed of eigenvalues of $M$, $v'$ is transformed vector after the spectral decomposition which will preserve the orthonormality: $\bar{v'} v=1$.
So what do you think of my derivation? Does minimization of the trace leads to the minimization of the expectation value?