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The rest of the exposition followed Lecture 4 of Trefethen and Bau pretty closely. In summary, the SVD of \(A \in \Cmn\) is
\begin{equation*}
AV = \Sigma U,
\end{equation*}
where
- \(U=(u_1 | u_2 | u_3 | \dots | u_m)
\text{,}\) the matrix that has as its columns the orthoronormal vectors in the direction of the principal semiaxes of the image of the unit sphere \(S \in \Rn
\text{.}\)
- \(\Sigma = \diag(\sigma_1, \sigma_2, \dots, \sigma_n)\) is the diagonal matrix containing the lengths of the principal semiaxes. We order them as
\begin{equation*}
\sigma_1 \geq \sigma_2 \geq \cdots \geq \sigma_n \geq
0.
\end{equation*}
- Then \(V=(v_1 | v_2 | v_3 | \dots | v_n)\) is the matrix whose columns are the preimages of the principal semiaxes, i.e., \(A v_i = \sigma_i u_i.\) It should be clear that the \(\{v_i\}\) form an orthonormal set, so \(V^{-1}=V^\star.\)
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