= Solution
By the <Eckart-Young-Mirsky theorem>, truncate the <singular value decomposition>:
$$
\boxed{A_r=\sum_{j=1}^r\sigma_j u_jv_j^T},\qquad
\boxed{\|A-A_r\|_2=\sigma_{r+1}},\qquad
\boxed{\|A-A_r\|_F=\left(\sum_{j=r+1}^n\sigma_j^2\right)^{1/2}}.
$$
The same $A_r$ minimizes the <spectral norm> and <Frobenius norm> errors over <matrices> of <matrix rank> at most $r$. This conventional meaning of rank-$r$ <low-rank approximation> includes the case $\operatorname{rank}A<r$, when $A_r=A$.
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