= Nonuniqueness of spectral-norm low-rank approximation
{title2=$\|A-(A_r+t u_1v_1^T)\|_2=\sigma_{r+1}\quad(|t|\leq\sigma_{r+1})$}
Even distinct <singular values> do not make a best <low-rank approximation> unique in the <spectral norm>. If $A=\sum_{j=1}^n\sigma_ju_jv_j^T$ has $\sigma_1>\cdots>\sigma_n>0$ and $1\leq r<n$, its <truncated singular value decomposition> $A_r$ can be changed to $A_r+t u_1v_1^T$ for any $|t|\leq\sigma_{r+1}$. Its <matrix rank> stays $r$ since $\sigma_1+t>0$, while the <spectral norm> error is $\max(|t|,\sigma_{r+1})=\sigma_{r+1}$, the optimum from the <Eckart-Young-Mirsky theorem>.
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