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lecture03 - Yinyu Ye MS&E Stanford Mathematical Foundations...

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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #02 1 Mathematical Foundations Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/˜yyye Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #02 2 Real n -Space; Euclidean Space R : real numbers R n : n -dimensional Euclidean space x y means x j y j for j = 1 , 2 , ..., n 0 : vector of all zeros (origin); e : vector/point of all ones inner-product or sum-product of two vectors: x y := x T y = n X j =1 x j y j Euclidean norm : k x k 2 = x T x , Infinity-norm : k x k = max {| x 1 | , | x 2 | , ..., | x n |} , p -norm : k x k p = n j =1 | x j | p · 1 /p
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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #02 3 Column vector/point : x = ( x 1 ; x 2 ; . . . ; x n ) and row vector : x = ( x 1 , x 2 , . . . , x n ) Transpose operation : x T A set of vectors a 1 , ..., a m is said to be linearly dependent if there are scalars λ 1 , ..., λ m , not all zero, such that the linear combination m X i =1 λ i a i = 0 A linearly independent set of vectors that span R n is a basis . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #02 4 Known Inequalities Cauchy-Schwarz : given x , y ∈ R n , | x T y | ≤ k x kk y k . Arithmetic-geometric mean : given x > 0 , x j n Y x j · 1 /n .
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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #02 5 Matrices Matrix : R m × n , i th row: a i. , j th column: a .j , ij th element: a ij All-zero matrix : 0 , and identity matrix : I Diagonal matrix : X = diag ( x ) Symmetric matrix : Q = Q T Positive Definite : Q ´ 0 iff x T Q x > 0 , for all x 6 = 0 Positive Semidefinite : Q 0 iff x T Q x 0 , for all x Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #02 6 Linear and Convex Combination When x and y are two distinct points in R n and α runs over R , { z : z = α x + (1 - α ) y } is the line
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