Editing Singular value decomposition (section)

===Low-rank matrix approximation===
Some practical applications need to solve the problem of [[Low-rank approximation|approximating]] a matrix {{tmath|\mathbf M}} with another matrix <math>\tilde{\mathbf{M}}</math>, said to be [[#Truncated SVD|truncated]], which has a specific rank {{tmath|r}}. In the case that the approximation is based on minimizing the [[Frobenius norm]] of the difference between {{tmath|\mathbf M}} and {{tmath|\tilde{\mathbf M} }} under the constraint that <math>\operatorname{rank}\bigl(\tilde{\mathbf{M}}\bigr) = r,</math> it turns out that the solution is given by the SVD of {{tmath|\mathbf M,}} namely

<math display=block>
\tilde{\mathbf{M}} = \mathbf{U} \tilde{\mathbf \Sigma} \mathbf{V}^*,
</math>

where <math>\tilde{\mathbf \Sigma}</math> is the same matrix as <math>\mathbf \Sigma</math> except that it contains only the {{tmath|r}} largest singular values (the other singular values are replaced by zero). This is known as the '''[[Low-rank approximation|Eckart–Young theorem]]''', as it was proved by those two authors in 1936 (although it was later found to have been known to earlier authors; see {{harvnb|Stewart|1993}}).