PCA formulation: Find the best rank-k subspace of d-dimensional spaceFinding the best rank-k subspace of...

PCA formulation: Find the best rank-k subspace of d-dimensional space
Finding the best rank-k subspace of d-dimensional space can be interpreted as the Principle Component Analysis (PCA) problem provided that the goal of optimization is minimizing the error in the sense of variance.
Assume we have access to the data sample $x_1,\dots ,x_n\in {\mathbb{R}}^n$, prove that the first problem is equivalent to the second one:
$\begin{array}{}\text{(1)}& \widehat{U_n}=\arg min\sum _{l=1}^n\Vert x_l-{\mathcal{P}}_U(x_l){\Vert }_2^2\end{array}$
such that
$U\in {\mathbb{R}}^{d\times k},U^{T}U=I_k$
$\begin{array}{}\text{(2)}& \widehat{U_n}=\arg min\Vert X_n-U{U}^T{X}_n{\Vert }_F^2\end{array}$
such that
$U\in {\mathbb{R}}^{d\times k},U^{T}U=I_k$
where Ik is Identity matrix, $P_U=U{U}^T$ is a projection matrix, $X_n=\left[\begin{array}{ccc}{x}_1& \cdots & x_n\end{array}\right]$ is a matrix of data, $\Vert \cdot {\Vert }_F$ is Frobenius norm, i.e. $\Vert A{\Vert }_F=\sqrt{\text{Trace}(A^{T}A)}$