Nonuniqueness of spectral-norm low-rank approximation ($\Vert A-(A_r+t{u}_1{v}_1^T){\Vert }_2={\sigma }_{r+1}(|t|\le {\sigma }_{r+1})$)

Even distinct singular values do not make a best low-rank approximation unique in the spectral norm. If $A={\sum }_{j=1}^n{\sigma }_j{u}_j{v}_j^T$ has ${\sigma }_1>\cdots >{\sigma }_n>0$ and $1\le r<n$, its truncated singular value decomposition $A_r$ can be changed to $A_r+t{u}_1{v}_1^T$ for any $|t|\le {\sigma }_{r+1}$. 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.