I'm trying to understand Singular Value Decomposition. So far I have seen the regular notation for the SVD of a matrix $A$ as \begin{equation} A=USV^T \end{equation} But there is a second representation of the SVD of A which is \begin{equation} A=\sum_{\alpha} s_{\alpha}\mathbf{u}_{\alpha} \mathbf{v}_{\alpha}^T \end{equation} However, I have not figured out how to go from the first to the second one or why it works. How can I prove the second one from the first one?
2026-03-29 03:03:55.1774753435
Proof of alternative writing of SVD
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