Theorem23.2

Let \(A_V\) be the rank-\(v\) approximation to \(A\)

\begin{equation*} A_v = \sum_{j=1}^r \sigma_j u_j v_j^\star. \end{equation*}

Then

\begin{equation*} \|A- A_v\|_2 = \inf_{\Rank(X)\leq v}\|A-X\|_2 = \sigma_{v+1}, \end{equation*}

where if \(v=p=\min(m,n)\text{,}\) and we define \(\sigma_{v+1}=0\text{.}\)

in-context