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# n6 - Atsushi Inoue ECG 765 Mathematical Methods for...

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Unformatted text preview: Atsushi Inoue ECG 765: Mathematical Methods for Economics Fall 2009 Lecture Notes 6 Vectors and Matrices To characterrize optimal solutions for multivariate objective functions, it is convenient to use vectors and matrices. Outline A. Vectors B. Matrices A. Vectors. Digression on the notation Σ x 1 + x 2 + x 3 = 3 X i =1 x j 7 X j =3 x j = x 3 + x 4 + x 5 + x 6 + x 7 n X k =0 x k = x + x 1 + ··· + x n 3 X i =1 ax i = ax 1 + ax 2 + ax 3 = a ( x 1 + x 2 + x 3 ) = a 3 X i =1 x i 3 X j =1 a j x j = a 1 x 1 + a 2 x 2 + a 3 x 3 n X k =0 a k x k = a x + a 1 x 1 + ··· + a n x n = a + a 1 x + ax 2 + ··· + a n x n 1 Definition (Inner Product). If x and y are elements of < n , we define the inner product , sometimes called the dot or scalar product , of x and y to be the real number < x , y > = ∑ n i =1 x i y i . Definition (Norm). The norm (or the length ) of x is defined be the real number k x k = < x , x > 1 2 = n X i =1 x 2 i ! 1 2 Example. The length of a two-dimensional vector x is k x k = q x 2 1 + x 2 2 . 6- x =(1,2) 2 1 k x k = √ 5 Theorem. Suppose x , y , z ∈ < n and α is a real number. Then (a) k x k ≥ (b) k x k = 0 if and only if x = (c) k α x k = | α |k x k (d) Cauchy-Schwarz inequality ) | < x , y > | ≤ k x kk y k (e) ( triangle inequality ) k x + y k ≤ k x k + k y k 2 Proofs. (a) Since x 2 i ≥ 0 for all i = 1 , 2 ,...,n , it follows that ∑ n i =1 x 2 i ≥ 0....
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n6 - Atsushi Inoue ECG 765 Mathematical Methods for...

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