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More precisely, it should be called as low-rank approximation.

Wiki and Horn P449 give good discussions on it. What I want to highlight here is the following assertion in wiki:

In order to minimize the above equation, it is said $U^*\tilde{M}V$ should be diagonal. This is easy to prove. If $U^*\tilde{M}V$ is not diagonal, we can always find another $\tilde{M}_2$ making $U^*\tilde{M}_2V$ be the diagonal of $U^*\tilde{M}V$, and hence giving smaller ||*||.

But when $U^*\tilde{M}V$ is diagonal, we can’t say $\tilde{M}=U\Sigma(\tilde{M})V$, right? Therefore, this proof is not rigorous.

In fact, Horn gave some theorems on this problem. See 7.4.50 and 7.4.51 on P447~448.

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