# Lower Rank Matrix

Given I have a matrix A of rank 3. I want to create a matrix of Rank 2 which is closest to A in the ${l}_{2}$ / Frobenius norm. Let's call this matrix F.

Is easy to achieve by the SVD, namely, if $A = U S {V}^{H}$ by the SVD decomposition $F = U \hat{S} {V}^{H}$. Where $\hat{S}$ is the same as $S$ with the last Singular Value zeroed.

The question is, is there a less computationally intensive method to create F but using the SVD decomposition?

Thanks.

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it's not necessary to compute the full SVD decomposition in order to obtain a low rank approximation. Instead, just compute the leading singular values and their corresponding left and right vectors. –  Victor May Feb 7 '12 at 8:37
There are ways, such as an adaptation of the power method, for computing partial singular value decompositions. You'll want to look into the literature for algorithms. –  Ｊ. Ｍ. Feb 7 '12 at 10:12
@J.M., Could you refer me to some sources? –  Drazick Feb 7 '12 at 13:23
Nash/Shlien, off the top of my head. There are probably more modern methods, but I don't have my notes with me at the moment... –  Ｊ. Ｍ. Feb 7 '12 at 13:30
Look for nipals PCA –  Donbeo Aug 14 at 14:39