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lecture03 - Yinyu Ye MS&E Stanford MS&E211 Lecture Note#03...

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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #03 1 Mathematical Foundations Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/˜yyye
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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #03 2 Real n -Space; Euclidean Space R : real numbers R n : n -dimensional Euclidean space x y means x j y j for all j = 1 , 2 , ..., n 0 : vector of all zeros (origin); 1 : vector/point of all ones inner-product or sum-product of two vectors: x y := x T y = n j =1 x j y j Euclidean norm : x 2 = x T x , Infinity-norm : x = max {| x 1 | , | x 2 | , ..., | x n |} , p -norm : x p = n j =1 | x j | p 1 /p
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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #03 3 vector/point : x = x 1 x 2 . . . x n or 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 i =1 λ i a i = 0 A linearly independent set of vectors that span R n is a basis .
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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #03 4 Known Inequalities Cauchy-Schwarz : given x , y ∈ R n , | x T y | ≤ x y . Triangle : given x , y ∈ R n , x + y x + y . Arithmetic-geometric mean : given x > 0 , x j n x j 1 /n .
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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #03 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 = 0 Positive Semidefinite : Q 0 iff x T Q x 0 , for all x
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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #03 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 entire line determined by x and y
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