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Techguru33 (talk | contribs) Tags: Mobile edit Mobile web edit |
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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.|292x292px]]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>\mathbf{A}</math>, can be efficiently represented by keeping the first <math>k</math> singular values and corresponding vectors. The truncated decomposition
<math>\mathbf{A}_k = \sum_{j=1}^k \sigma_j\mathbf{u}_j \mathbf{v}_j^T </math>
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