Slides_2010_02_01

# Slides_2010_02_01 - Applied linear algebra and numerical...

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Unformatted text preview: Applied linear algebra and numerical analysis Session 12 Prof. Ulrich Hetmaniuk Department of Applied Mathematics January 31, 2010 Norms Definition Consider U a real linear space of elements . A norm k·k is a function from U → R + , satisfying the following conditions : ∀ u ∈ U , k u k ≥ , (1a) k u k = 0 if and only if u = ¯ , (1b) ∀ α ∈ R , k α u k = | α |k u k , (1c) ∀ u ( 1 ) , u ( 2 ) ∈ U , u ( 1 ) + u ( 2 ) ≤ u ( 1 ) + u ( 2 ) (1d) Example of a norm for functions Example Consider U = C ([ , 1 ] , R ) . Show that the map C ([ , 1 ] , R )-→ R + f 7-→ q ´ 1 f ( t ) 2 dt is a norm. The proof uses the Hölder inequality ˆ 1 f ( t ) g ( t ) dt ≤ s ˆ 1 f ( t ) 2 dt s ˆ 1 g ( t ) 2 dt Norm of a Matrix Example The Frobenius norm of real matrices in R m × n k A k F = s m ∑ i = 1 n ∑ j = 1 | a ij | 2 , (2) Example The induced matrix 2-norm is defined as k A k 2 = max x 6 = k Ax k 2 k x k 2 = max k x k 2 = 1 k Ax k 2 . (3) Norm of a Matrix • k A k 2 = max x 6 = k Ax k 2 k x k 2 = max x q x T A T Ax x T x • k A k 1 = max x 6 = k Ax k 1 k x k 1 = max j ( ∑ m i = 1 | a ij | ) • It is the maximum sum among the column vectors of A . • k A k ∞ = max x 6 = k Ax k ∞ k x k ∞ = max i ( ∑ n j = 1 | a ij | ) • It is the maximum sum among all the rows of A . • Note k A k 1 = A T ∞ • Remember ( p = 1 , 2 , ∞ ) k Ax k p ≤ k A k p k x k p k AB k p ≤ k A k p k B k p Inner Product Definition Consider U a real linear space. An inner product < · , · > is a map U × U → R satisfying the following properties ∀ u ∈ U ,< u , u > ≥ , (4a) <...
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Slides_2010_02_01 - Applied linear algebra and numerical...

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