Ans 4 - v2

Ans 4 - v2 - LECTURE 3 Norms The Euclidean length is...

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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 2-norm , 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 1-norm is the sum of magnitudes of the ele- ments of x : x 1 = n i =1 | x i | . 3-1 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 p-norms defined by x p = n i =1 x p i 1 /p 1 ≤ p ≤ ∞ . 3-2 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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Ans 4 - v2 - LECTURE 3 Norms The Euclidean length is...

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