Singular Value Decomposition Properties

Let $X_{p \times n}$ be a data matrix with row means 0. Let $L_{(j)}$ denote the $j$th left singular vector of the singular value decomposition of $X = UDV^T$. Let $G_{(j)} = L_{(j)}^T X$. Then how do you go about calculating $\hat{X}$, (the projection of $X$ into $\mathcal{L}_{row}(G_{j})$), and $(X-\hat{X})(X-\hat{X})^T$?

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Could you possibly clarify what $Y_j$ is? –  icurays1 Nov 14 '12 at 6:02
excuse me, that should have been a G –  sheten Nov 14 '12 at 7:01
What is $\mathcal{L}_{row}(G_{j})$? –  littleO Nov 14 '12 at 8:48
The row space of $G_(j)$ –  sheten Nov 14 '12 at 15:03