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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 deﬁned as κ ( A ) = k A k ² ² A1 ² ² . Show that for any invertible matrix A , κ ( A ) ≥ 1. [ Hint: Apply the multiplication property to k I k = ² ² AA1 ² ² , 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 xy k ). Show that g ( y ) = g ( x )+ ∇ g ( x )( yx )+ Z 1 [ ∇ g ( x + s ( yx ))∇ g ( x )]( yx ) ds . Use this to show that k g ( y )[ g ( x )+ ∇ g ( x )( yx )] k ≤ 1 2 L k yx k 2 . 2...
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 Spring '12
 DavidStewart
 Linear Algebra, 1 g, 1 g, κ, 22m

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