cs357-slides-norms - Norms L Olson October 1 2015...

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Norms L. Olson October 1, 2015 Department of Computer Science University of Illinois at Urbana-Champaign 1
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objectives Set up an array of data and measure its “size” Construct a “norm” and apply its properties to a problem Describe a “matrix norm” or “operator norm” Find examples where a matrix norm is appropriate and not appropriate 2
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vector addition and subtraction Addition and subtraction are element-by-element operations c = a + b ⇐⇒ c i = a i + b i i = 1 , . . . , n d = a - b ⇐⇒ d i = a i - b i i = 1 , . . . , n a = 1 2 3 b = 3 2 1 a + b = 4 4 4 a - b = - 2 0 2 3
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multiplication by a scalar Multiplication by a scalar involves multiplying each element in the vector by the scalar: b = σ a ⇐⇒ b i = σ a i i = 1 , . . . , n a = 4 6 8 b = a 2 = 2 3 4 4
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vector transpose The transpose of a row vector is a column vector: u = 1 , 2 , 3 then u T = 1 2 3 Likewise if v is the column vector v = 4 5 6 then v T = 4 , 5 , 6 5
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linear combinations Combine scalar multiplication with addition α u 1 u 2 . . . u m + β v 1 v 2 . . . v m = α u 1 + β v 1 α u 2 + β v 2 . . . α u m + β v m = w 1 w 2 . . . w m r = - 2 1 3 s = 1 0 3 t = 2 r + 3 s = - 4 2 6 + 3 0 9 = - 1 2 15 6
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linear combinations Any one vector can be created from an infinite combination of other “suitable” vectors. w = " 4 2 # = 4 " 1 0 # + 2 " 0 1 # w = 6 " 1 0 # - 2 " 1 - 1 # w = " 2 4 # - 2 " - 1 1 # w = 2 " 4 2 # - 4 " 1 0 # - 2 " 0 1 # 7
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linear combinations Graphical interpretation: Vector tails can be moved to convenient locations Magnitude and direction of vectors is preserved [1,0] [0,1] [2,4] [1,-1] [4,2] [-1,1] [1,1] 0 1 2 3 4 5 6 0 1 2 3 4 8
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vector inner product In physics, analytical geometry, and engineering, the dot product has a geometric interpretation σ = x · y ⇐⇒ σ = n X i = 1 x i y i x · y = k x k 2 k y k 2 cos θ 9
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vector inner product The inner product of x and y requires that x be a row vector y be a column vector h x 1 x 2 x 3 x 4 i y 1 y 2 y 3 y 4 = x 1 y 1 + x 2 y 2 + x 3 y 3 + x 4 y 4 10
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vector inner product For two n -element column vectors, u and v , the inner product is σ = u T v ⇐⇒ σ = n X i = 1 u i v i The inner product is commutative so that (for two column vectors) u T v = v T u 11
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vector outer product The inner product results in a scalar. The outer product creates a rank-one matrix: A = uv T ⇐⇒ a i , j = u i v j Example Outer product of two 4-element column vectors uv T = u 1 u 2 u 3 u 4 h v 1 v 2 v 3 v 4 i = u 1 v 1 u 1 v 2 u 1 v 3 u 1 v 4 u 2 v 1 u 2 v 2 u 2 v 3 u 2 v 4 u v u v u v u v 12
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vector norms Compare magnitude of scalars with the absolute value α > β
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