E i i 11 17 singular values the singular values of a

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Unformatted text preview: σ y with σ = A 2 . There exists V2 ∈ IRn×(n−1) and U2 ∈ IRm×(m−1) so V = [x V2 ] ∈ IRn×n and U = [y U2 ] ∈ IRm×m are orthogonal. It follows that σw σ y AV2 = U AV = U ≡ A1 0B for some w and B . Since A1 σ w 2 ≥ (σ 2 + w w)2 2 we have A1 2 ≥ (σ 2 + w w). But σ 2 = A 2 be 0. By induction, we complete this proof. 2 2 = A1 2 , and thus w must 2 10 / 17 Singular vectors As A = U ΣV , by comparing the columns in the equations AV = U Σ and A U = V Σ , it is easy to show Avi A ui = σ ui = σ vi where i = 1, . . . , min(m, n) The σi are the singular values of A and the vectors ui and vi are the i -th left singular vector and i -th right singular vector respectively U is a set of eigenvectors of AA ∈ IRm×m Σ is a diagonal matrix whose values are the square root of eigenvalues of AA ∈ IRm×m V is a set of eigenvectors of A A ∈ IRn×n It can be shown that singular values σi are the square roots of √ eigenvalues, λi , i.e., σi = λi 11 / 17 Singular values The singular values of a matrix A are precisely the lengths of the semi-axes of the hyperellipsoid E defined by E = {Ax : x 2 = 1} The semi-axes are described by the singular vectors The SVD reveals the struct...
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This document was uploaded on 02/10/2014.

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