For a matrix H of size $M\times N$ with zero mean random entries with variance $\beta$, I am trying to find $$\mathbb{E}\{HVV^H H^H\}=\beta tr(VV^H) I_M,$$where V is of size $N\times d$ with $d<N$.
The problem is that the rank of left side is $d$, whereas right side has the rank of $M$.
Can someone inform me the correct way to get the value of the expectation?