An equality in DOA estimation
I think I know the answer now. If I made an error, please correct me.

Assume there are $m$ sensors and $d$ sources. As the eigenvectors in $S$ and $G$ of the Hermitian matrix $R$ is selected to be orthogonal and normalized, 
$$UU^*=\begin{bmatrix}S&amp;G\end{bmatrix}\begin{bmatrix}S^*\\G^*\end{bmatrix}=SS^*+GG^*=UU^{-1}=I_m;$$
$$S^*S=I_d;$$
$$\mathrm{span}\{G\}=\mathrm{span}\{S\}^{\bot}.$$

We know the noise subspace $\mathrm{span}\{G\}$ and $\mathrm{span}\{A\}$ are orthogonal as in the MUSIC algorithm:
$$\mathrm{span}\{G\}=\mathrm{span}\{A\}^{\bot}.$$
So $\mathrm{span}\{A\}=\mathrm{span}\{S\}$, which implies their projection matrices into this subspace are equal:
$$A(A^*A)^{-1}A^*=S(S^*S)^{-1}S^*.$$
Using $S^*S=I_d$, we get
$$A(A^*A)^{-1}A^*=SS^*;$$
with $SS^*+GG^*=I_m$, we get
$$GG^*=I_m-SS^*=I_m-A(A^*A)^{-1}A^*.$$