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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}}).
=== Image compression ===
[[File:Svd compression.jpg|thumb|Singular-value decomposition (SVD) image compression of a 1996 Chevrolet Corvette photograph. The original RGB image (upper-left) is compared with rank 1, 10, and 100 reconstructions.|280x280px]]One practical consequence of the low-rank approximation given by SVD is that a greyscale image represented as an <math>m \times n</math> matrix <math>A</math>, can be efficiently represented by keeping the first <math>k</math> singular values and corresponding vectors. The truncated decomposition
<math>A_k = \mathbf{U}_k \mathbf{\Sigma}_k \mathbf{V}^T_k</math>
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