If the expectation $\langle v,Mv \rangle$ of an operator is $0$ for all $v$ is the operator $0$?
Solution 1:
This is true in general for all vector spaces over $\mathbb C$. Indeed, one can consider $v = x-y$ and $v=x-iy$ to come up with the identity $\langle x, My \rangle=0$ for all $x, y$.