# Let x irn and y irm be unit 2 norm vectors that

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Unformatted text preview: , u2 , . . . , un } oriented in the directors of the principal semi-axes of AS The n right singular vectors of A are the unit vectors {v1 , v2 , . . . , vn } ∈ S that are the preimages of the principal semi-axes of AS , numbered so that Avj = σj uj  ¤©¥ ¦¨§ ©¥ ¦ £ ¥£   i ¥¡¥ ¦¤¡¢ ¥£ Singular values and singular vectors Singular value decomposition (SVD) Theorem If A is a real m-by-n matrix, then there exists orthogonal matrices U = [u1 , . . . , um ] ∈ IRm×m , and V = [v1 , . . . , vn ] ∈ IRn×n such that U AV = Σ = diag(σ1 , . . . , σp ) ∈ IRm×n , p = min(m, n), where σ1 ≥ σ2 ≥ . . . ≥ σp ≥ 0, or A = U ΣV The σi are the singular values of A and the vectors ui and vi are the i -th left singular vector and the i -th right singular vector respectively It follows that AV = U Σ, and A U = V Σ 9 / 17 Existence of SVD Proof. Let x ∈ IRn and y ∈ IRm be unit 2-norm vectors that satisfy Ax =...
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## This document was uploaded on 02/10/2014.

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