# r 0 0v for any with 0 r a 1 i ui

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Unformatted text preview: corresponding to σ1 Once σ1 , v1 , and v1 are determined, the remainder of SVD is determined by the action of A on the space orthogonal to v1 Since v1 is unique up to a sign, the orthogonal space is unique deﬁned and so are the remaining singular values 5 / 20 Matrix properties via SVD Theorem The rank of A is r , the number of nonzero singular values. Proof. The rank of a diagonal matrix is equal to the number of its nonzero entries, and in SVD, A = U ΣV where U and V are of full rank. Thus, rank(A) = rank(Σ) = r Theorem A 2 = σ1 , and A F = 2 2 σ1 + · · · + σr Proof. As U and V are orthogonal, A = U ΣV , A 2 = Σ 2 . By deﬁnition, Σ 2 = max x =1 Σx 2 = max{|σi |} = σ1 . Likewise, A F = Σ F , and by deﬁnition Σ F = 2 2 σ1 + · · · + σr 6 / 20 Eigenvalue decomposition From linear algebra, Ax = λx, λ is an eigenvalue, and x is an eigenvector For m eigenvectors, λ1 λ2 A[x1 , x2 , . . . , xm ] = [x1 , x2 , . . . , xm ] .. . λm and AX = X Λ where Λ is an m × m diagonal matrix whose entries are the eigenvalues of A, and X ∈ IRm×m contains linearly independent eigenvector of A The eigenvalue decomposition of A A = X ΛX −1 7 / 20 SVD and eigenvalue decomposition SVD uses two diﬀerent bases (the sets of left and right singular vectors), whereas the eigenvalue decomposition uses just one (eigenvectors) SVD uses orthonormal bases, whereas the eigenvalue decomposition uses a basis that generically is not orthogonal Not all matrices have an eig...
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