# proof of Theorem 1

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Let $\textbf{u}_k$ and $\textbf{v}_k$ denote column k of $\textbf{U}$ and $\textbf{V}$ respectively, We prove the theorem by expanding out the squared Frobenius norm and rearranging terms:

\begin{align} \| \textbf{X} - \textbf{U}\textbf{D}\textbf{V}^T \|^2_F & = tr((\textbf{X} - \textbf{U}\textbf{D}\textbf{V}^T)^T(\textbf{X} - \textbf{U}\textbf{D}\textbf{V}^T)) \\ & = -2tr(\textbf{V}\textbf{D}\textbf{U}^T\textbf{X}) + tr(\textbf{V}\textbf{D}\textbf{U}^T\textbf{U}\textbf{D}\textbf{V}^T) + \|\textbf{X}\|^2_F \\ & = \sum_{k=1}^K d_k^2 - 2tr(\textbf{V}\textbf{D}\textbf{U}^T\textbf{X}) + \|\textbf{X}\|^2_F \\ & = \sum_{k=1}^K d_k^2 - \sum_{k=1}^K d_k\textbf{u}_k^T\textbf{X}\textbf{v}_k + \|\textbf{X}\|^2_F \end{align}

The above proof is the proof of Theorem 2.1 in <ref name="WTH2009">Daniela M. Witten, Robert Tibshirani, and Trevor Hastie. (2009) "A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis". Biostatistics, 10(3):515–534.</ref>

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