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 kk 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) R , k u k = | |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 Hlder inequality 1 f ( t ) g ( t ) dt s 1 f ( t ) 2 dt s 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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