induced by the , vector norm is defined as For any 1 < p < x,...

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induced by the , vector norm is defined as For any 1 < p < x, the operator norm || Az||p ||A| = sup A is equal to the maximum column sum. We have seen that ||A||1 = maxje[n] (a). Prove that | Alo = maxie[m] Σj1 Ayl (b). Prove that ||A||2= Amax (ATA). Furthermore, recall that any matrix A € IR admits a singular value decomposition (SVD) Rank(A) (=1 A= a(A)uvt. where Rank(A) is the rank of A. (o(A), = 1,..., Rank(A)) are the singular values satisfying 01(A) 2 02(A) 2... 2 Rank(A) (A), and u. v, are the singular vectors in IR" and IR" respectively. Using SVD of A, prove that ||A|201(A). (Hint: for any symmetric matrix M, Amax (M)= max 0.) induced by the , vector norm is defined as For any 1 < p < x, the operator norm || Az||p ||A| = sup A is equal to the maximum column sum. We have seen that ||A||1 = maxje[n] (a). Prove that | Alo = maxie[m] Σj1 Ayl (b). Prove that ||A||2= Amax (ATA). Furthermore, recall that any matrix A € IR admits a singular value decomposition (SVD) Rank(A) (=1 A= a(A)uvt. where Rank(A) is the rank of A. (o(A), = 1,..., Rank(A)) are the singular values satisfying 01(A) 2 02(A) 2... 2 Rank(A) (A), and u. v, are the singular vectors in IR" and IR" respectively. Using SVD of A, prove that ||A|201(A). (Hint: for any symmetric matrix M, Amax (M)= max 0.)