[Zhan Xiang matrix theory exercise reference] exercise 4.11

11. $ M_n $ the norm on $ \ Sen {\ cdot} $ is symmetric, if $ \ Bex \ Sen {ABC} \ Leq \ Sen {A }_\ infty \ Sen {C }_\ infty \ Sen {B }, \ quad \ forall \ A, B, C \ In M_n. \ EEx $ proof: $ \ Sen {\ cdot} $ symmetric when and only when $ \ Sen {\ cdot} $ is unchanged.

 

 

 

Proof: $ \ rA $: $ u, v $, $ \ beex \ Bea \ Sen {UAV} & \ Leq \ Sen {u }_\ infty \ Sen {v }_\ infty \ Sen {A }=\ Sen {}; \\\ Sen {A} <=\ Sen {u ^ * (UAV) V ^ *} \ Leq \ Sen {u ^ *} _ \ infty \ Sen {V ^ * }_\ infty \ Sen {UAV }=\ Sen {UAV }. \ EEA \ eeex $ \ Bex \ Sen {UAV} =\sen {A}, \ quad \ forall \ A \ In M_n. \ EEx $ \ la $: evidence: $ \ Bex \ Sen {ABC} \ Leq \ Sen {A }_\ infty \ Sen {C }_\ infty B }, \ EEx $ the theorem dominated by the fan (Theorem 4.24), which must only prove $ \ Bex S (ABC) \ prec_w S (\ Sen {A }_\ infty \ Sen {C }_\ infty B ), \ EEx $, which can be $ \ beex \ Bea s_j (ABC) & \ Leq S_1 (a) s_j (BC) \ quad \ sex {\ mbox {inference 4.3 }\\\& \ Leq S_1 (a) s_j (B) S_1 (c) \ quad \ sex {\ mbox {inference 4.3 }\\\&=\ Sen {A }_\ infty \ Sen {C }_\ infty s_j (B) \ quad \ sex {S_1 () =\sen {A }_\ infty }\\\& = s_j (\ Sen {A }_\ infty \ Sen {C }_\ infty B) \ EEA \ eeex $ release now.

[Zhan Xiang matrix theory exercise reference] exercise 4.11