[Zhan Xiang matrix theory exercise reference] exercise 3.4

4. Set the components $ X, Y, U \ In \ BBR ^ N $ to decrease. proof:

 

(1). If $ x \ prec y $, then $ \ SEF {x, u} \ Leq \ SEF {Y, u} $.

 

(2 ). if $ x \ prec_w y $ and $ U \ In \ BBR ^ N _ + $, then $ \ SEF {x, u} \ Leq \ SEF {Y, u} $.

 

 

Proof:

 

(1 ). note by $ x \ prec y $ \ Bex S_k = \ sum _ {I = 1} ^ K X_ I, \ quad t_l = \ sum _ {j = 1} ^ L y_l, \ EEx $ then $ \ bee \ label {3_4_decay} S_k \ Leq t_k, \ quad k = 1, \ cdots, n-1; \ quad s_n = t_n. \ EEE $ then $ \ beex \ Bea \ SEF {X, u} & = \ sum _ {I = 1} ^ n x_iu_ I \ & = s_1u_1 + \ sum _ {I = 2} ^ N (s_i-s _ {I-1 }) u_ I \ & = s_1u_1 + \ sum _ {I = 2} ^ n s_iu_ I-\ sum _ {I = 1} ^ {n-1} s_iu _ {I + 1 }\\ & =\ sum _ {I = 1} ^ n s_iu_ I-\ sum _ {I = 1} ^ {n-1} s_iu _ {I + 1} \ & = \ sum _ {I = 1} ^ {n-1} s_ I (u_i-u _ {I + 1 }) + s_nu_n \ & \ Leq \ sum _ {I = 1} ^ {n-1} t_ I (u_i-u _ {I + 1 }) + t_nu_n \ quad \ sex {\ eqref {3_4_decay }}\\\&=\ SEF {Y, u }. \ EEA \ eeex $

 

(2 ). mark same as above, with $ \ beex \ Bea \ SEF {x, u} & =\ sum _ {I = 1} ^ {n-1} s_ I (u_i-u _ {I-1 }) + s_nu_n \ & \ Leq \ sum _ {I = 1} ^ {n-1} t_ I (u_i-u _ {I + 1 }) + t_nu_n \ & \ quad \ sex {s_ I \ Leq t_ I, \ I = 1, \ cdots, n-1; \ s_n \ Leq t_n, u_n \ geq 0 }\\&=\ SEF {Y, u }. \ EEA \ eeex $

[Zhan Xiang matrix theory exercise reference] exercise 3.4