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Unformatted text preview: LECTURE 3 Norms The Euclidean length is defined as x 2 = √ x T x = ( x 2 1 + x 2 2 + · · · + x 2 n ) 1 / 2 = n i =1 x 2 i 1 2 . This is the l 2norm , and it is the usual way for measuring a vector’s length. There are others: • The l ∞norm is the magnitude of the largest ele ment in x : x ∞ = max 1 ≤ i ≤ n  x i  . • The l 1norm is the sum of magnitudes of the ele ments of x : x 1 = n i =1  x i  . 31 Definition A norm is a function · : R m → R that satisfies 1. x ≥ 0, and x = 0 only if x = 0 2. x + y ≤ x + y (triangle inequality) 3. αx =  α  x for all x, y ∈ R n and all α ∈ R . The three norms introduced above ( l 1 , l 2 and l ∞ ) are all special cases of a family of pnorms defined by x p = n i =1 x p i 1 /p 1 ≤ p ≤ ∞ . 32 Suppose we want to measure the distance between x = 11 12 13 and y = 12 14 16 . Let z = y x = 1 2 3 . Then z 1 = 1 + 2 + 3 = 6 z 2 = √ 1 + 4 + 9 = √ 14 z ∞ = 3 In Matlab we get the above results by typing z = [12,14,16]’  [11,12,13]’; norm(z,1); norm(z); norm(z,inf) In this example z 1 ≥ z 2 ≥ z ∞ ....
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This note was uploaded on 01/14/2011 for the course ECE 210a taught by Professor Chandrasekara during the Fall '08 term at UCSB.
 Fall '08
 Chandrasekara

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