Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 13J / c

Independence gives $\mathbb{E}\Vert Y^{\ast }-HY{\Vert }^2=\Vert (I-H)\mu {\Vert }^2+n{\sigma }^2+{\sigma }^2\mathrm{tr}{H}^2$. Comparing with part (b) shows the difference is $2{\sigma }^2\mathrm{tr}H$, proving unbiasedness.