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X a1 f f x and the taylor series expansion x x x0

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Unformatted text preview: envalue decomposition, but all matrices have a SVD 8 / 20 Matrix properties via SVD (cont’d) Theorem The nonzero singular values of A are the square roots of the nonzero eigenvalues of AA or A A (they have the same nonzero eigenvalues). Proof. From definition, AA = (U ΣV )(U ΣV ) = U ΣV V ΣU 2 2 = U diag(σ1 , . . . , σp ) U Theorem For A ∈ IRm×m , | det(A)| = m i =1 σi Proof. m |det(A)| = |det(U ΣV )| = |det(U )||det(Σ)||det(V )| = |det(Σ)| = σi i =1 9 / 20 Low-rank approximation Theorem (Eckart-Young 1936) Let A = U ΣV = U diag(σ1 , . . . , σr , 0, . . . , 0)V . For any ν with 0 ≤ ν ≤ r , Aν = ν=1 σi ui vi , i A − Aν 2 = min A − B 2 = σν +1 rank(B )≤ν Proof. Suppose there is some B with rank(B ) ≤ ν such that A − B 2 < A − Aν 2 = σν +1 . Then there exists an (n − ν )-dimensional subspace W ∈ IRn such that w ∈ W ⇒ B w = 0. Then Aw 2 = (A − B )w 2 ≤ A − B 2 w 2 < σν +1 w 2 Thus W is a (n − ν )-dimensional subspace where Aw < σν +1 w . But there is a (ν + 1)-dimensional subspace where Aw ≥ σν +1 w , namely the space spanned by the first ν + 1 right singular vector of A. Since the sum of the dimensions of these two spaces exceeds n, there must be a nonzero vector lying in both, and this is a contradiction. 10 / 20 Low-rank approximat...
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