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Unformatted text preview: erties that k A x k k A kk x k and k AB k k A kk B k . If the vector norm is k x k : = max i | x i | , show that the corresponding matrix norm is k A k = max i j a ij . 4. The condition number of an invertible matrix A is dened as ( A ) = k A k A-1 . Show that for any invertible matrix A , ( A ) 1. [ Hint: Apply the multipli-cation property to k I k = AA-1 , and check that k I k = 1.] 1 5. Suppose g : R n R n has a Lipschitz continuous Jacobian matrix ( k g ( x )- g ( y ) k L k x-y k ). Show that g ( y ) = g ( x )+ g ( x )( y-x )+ Z 1 [ g ( x + s ( y-x ))- g ( x )]( y-x ) ds . Use this to show that k g ( y )-[ g ( x )+ g ( x )( y-x )] k 1 2 L k y-x k 2 . 2...
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- Spring '12