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Let's first compute$A^T{\stackrel{藱}{u}}_i.$

$\begin{array}{rcl}{A}^T{\stackrel{藱}{u}}_i& =& \left(U危V^T\right){\stackrel{藱}{u}}_i\\ & =& \left(V{危}^T{U}^T\right){\stackrel{藱}{u}}_i\\ & =& V{危}^T\left(U^T{\stackrel{藱}{u}}_i\right)\\ & =& V{危}^T{e}_i\\ & & \end{array}$

where${\mathbf{e}}_i$ is the$i^{\text{th}}$ basis vector of${\mathrm{鈩�}}^n.$

Since${危}^T$ is an $n脳m$matrix

role="math" localid="1660679956475" ${危}^T{\mathbf{e}}_i=\left\{\begin{array}{ll}{蟽}_i{\mathbf{e}}_i& \text{if}1猢�i猢�m\\ 0& \text{otherwise}\end{array}\right.$

Hence,

role="math" localid="1660680127331" $V{危}^T{\mathbf{e}}_i=\left\{\begin{array}{ll}V{蟽}_i{\mathbf{e}}_i\mathbf{}\mathbf{=}{\mathbf{蟽}}_{\mathbf{i}}{\stackrel{\mathbf{藱}}{\mathbf{v}}}_{\mathbf{i}}& \text{if}1猢�i猢�m\\ 0& \text{otherwise}\end{array}\right.$

Thus,

$A^T{\stackrel{藱}{u}}_i=\left\{\begin{array}{ll}{\mathbf{蟽}}_{\mathbf{i}}{\stackrel{\mathbf{藱}}{\mathbf{v}}}_{\mathbf{i}}& \text{if}1猢�i猢�m\\ 0& \text{otherwise}\end{array}\right.$

Recall that${\stackrel{藱}{v}}_i\text{'}s$are eigenvectors of$A^{T}A$with eigenvalues${蟽}_i^2.$

Hence, $\left(A^{T}A\right){\stackrel{藱}{v}}_i={蟽}_i^2{\stackrel{藱}{v}}_i.$This implies that${\left(A^{T}A\right)}^{-1}\left({蟽}_i{\stackrel{藱}{v}}_i\right)=\frac{1}{{蟽}_i}{\stackrel{藱}{v}}_i.$

This implies the following.

${\left(A^{T}A\right)}^{-1}{A}^T{\stackrel{藱}{u}}_i=\left\{\begin{array}{ll}\frac{1}{{蟽}_i}{\stackrel{藱}{v}}_i& \text{if}1猢�i猢�m\\ 0& \text{otherwise}\end{array}\right.$

Now, use that

$A{\stackrel{藱}{v}}_i={蟽}_i{\stackrel{藱}{u}}_i.$

Hence,

$A{\left(A^{T}A\right)}^{-1}{A}^T{\stackrel{藱}{u}}_i=\left\{\begin{array}{ll}{\stackrel{藱}{u}}_i& \text{if}1猢�i猢�m\\ 0& \text{otherwise}\end{array}\right.$

If we are considering the sub space of, say W, which is spanned by$\left\{{\stackrel{藱}{u}}_1,{\stackrel{藱}{u}}_2,鈰�,{\stackrel{藱}{u}}_m\right\}$, then the above result implies that the projection ${\stackrel{藱}{u}}_i$onto W is ${\stackrel{藱}{u}}_i$if $1猢�i猢�m$and is 0 if $m+1猢�i猢�n.$

$A{\left(A^{T}A\right)}^{-1}{A}^T{\stackrel{藱}{u}}_i=\left\{\begin{array}{ll}{\stackrel{藱}{u}}_i& \text{if}1猢�i猢�m\\ 0& \text{otherwise}\end{array}\right.$