lecture03

lecture03 - Yinyu Ye, MS&E, Stanford...

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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 1 ,..., m ,notallzero,suchthatthe linearcombination m X i =1 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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lecture03 - Yinyu Ye, MS&E, Stanford...

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