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Why is it so important? Are there any applications which have a real impact?

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A lot of the impact is compuational - once you have the SVD, you can quickly compute the pseudo-inverse of a matrix, for example. en.wikipedia.org/wiki/… –  Thomas Andrews Dec 28 '12 at 15:04
    
Latent semantic indexing is used in recommendation systems, such as Netflix, and is based on SVD for the purpose of dimensionality reduction so that similarity measures can be calculated on the smaller data set to produce recommendations. –  lewellen Dec 28 '12 at 20:09
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