From what I've observed in practice the SVD gives the best rank 1 approximation with respect to the Frobenius norm. But from what I've heard from others, it also minimizes the distance to the L2 matrix norm. I'm wondering doesSVD give the best rank 1 approximation with respect to the Frobenius norm, L2 norm, or both?
2026-03-25 07:44:47.1774424687
Does SVD give the best rank 1 approximation with respect to the Frobenius norm, L2 norm, or both?
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The SVD produces the best rank-1 approximation (or more generally, the best approximation of at most any fixed rank) with respect to both the Frobenius and $L_2$ norms. Proofs of these results are given on this wiki page.