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Y xTy st jjyjj 2 1 CauchySchwarz implies that xTy jjxjjjjyjj jjxjj and y= x jjxjj achieves this bound Proof of (3) We have jjxjj 1 =max y xTy st jjyjj 1 1 So y opt= sign(x) and the optimal value is jjxjj 1 2 Positive semide nite matrices We denote by S n the set of all symmetric (real) n nmatrices 21 De nition De nition 6 A matrix(b) T R2!R2, with T x y = x2 y2 Solution This is NOT a linear transformation It can be checked that neither property (1) nor property (2) from above hold Let's show thatV W X Y Z \ ^ _ ` a b c d e f g h i j k l m n o p q r s t u v Y X ` W f w x ^ Z a m k i \ u y ^ ` m k X Y Z _ W x w a o i \ q l z ê




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