Denote SVD of $A$ by $A=USV^T$ where $U\in \mathbb{R}^{d_1 \times d_1}$ and $V\in \mathbb{R}^{d_2 \times d_2}$. For a given $r \in [\min\{d_1,d_2\}]$, let $U_r = [u_1,...,u_r]$ and $V_r = [v_1,...,v_r]$. Define $T$ to be the subspace spanned by all matrices in $\mathbb{R^{d_1 \times d_2}}$ of the form $U_rA^T$ or $BV^T_r$ for any $A \in \mathbb{R}^{d_2\times r}$ or $B \in \mathbb{R}^{d_1 \times r}$.
Prove that the orthogonal projection of any matrix $M \in \mathbb{R}^{d_1 \times d_2}$ is given by:
$$ P_T(M) = U_r U^T_r M + MV_rV^T_r - U_r U^T_r M V_r V^T_r $$
Mentioned (Equation 3.5) without proof in this paper.