lecture03

lecture03 - Yinyu Ye MS&E Stanford MS&E211 Lecture...

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Unformatted text preview: Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #02 1 MathematicalFoundations YinyuYe DepartmentofManagementScienceandEngineering StanfordUniversity Stanford,CA94305,U.S.A. http://www.stanford.edu/˜yyye Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #02 2 Real n-Space;EuclideanSpace • R :realnumbers • R n : n-dimensional Euclideanspace • x ≥ y means x j ≥ y j for j =1 , 2 ,...,n • :vectorofallzeros(origin); e :vector/pointofallones • inner-productorsum-product oftwovectors: x • y := x T y = n X j =1 x j y j • Euclideannorm : 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 Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #02 3 • Columnvector/point : x =( x 1 ; x 2 ; ... ; x n ) and rowvector : x =( x 1 ,x 2 ,...,x n ) • Transposeoperation : x T • Asetofvectors a 1 ,..., a m issaidtobe linearlydependent ifthereare scalars λ 1 ,...,λ m ,notallzero,suchthatthe linearcombination m X i =1 λ i a i = • A linearlyindependent setofvectorsthatspan R n isa basis . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #02 4 KnownInequalities • Cauchy-Schwarz :given x , y ∈R n , | x T y |≤k x kk y k . • Arithmetic-geometricmean :given x > , ∑ x j n ≥ ‡ Y x j · 1 /n . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #02 5 Matrices • Matrix : R m × n , i throw: a i. , j thcolumn: a .j , ij thelement: a ij • All-zeromatrix : ,and identitymatrix : I • Diagonalmatrix : X = diag ( x ) • Symmetricmatrix : Q = Q T • PositiveDefinite : Q ´ iff x T Q x > , forall x 6 = • PositiveSemidefinite : Q ” iff x T Q x ≥ , forall x Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #02...
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This note was uploaded on 05/06/2010 for the course MSE 211 at Stanford.

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lecture03 - Yinyu Ye MS&E Stanford MS&E211 Lecture...

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