15. (fan-Hoffman) set $ A, H \ In M_n $, where $ h $ is the Hermite matrix, then $ \ Bex \ Sen {A-\ re a} \ Leq \ Sen {A-H} \ EEx $ is set to any Unio constant norm.
Proof: (1 ). first, it is proved that $ \ Bex \ Sen {\ cdot} \ mbox {is an undo norm }, X \ In M_n \ Ra \ Sen {x }=\ Sen {x ^ *}. \ EEx $ in fact, $ x $ has the same singular value as $ x ^ * $, while $ \ Bex S (x) \ prec S (x ^ *) \ prec S (X ). \ EEx $ the fan-dominated principle, $ \ Bex \ Sen {x} \ Leq \ Sen {x ^ *} \ Leq \ Sen {x }. \ EEx $(2 ). original Certificate question. by $ \ beex \ Bea-\ re a & = A-\ frac {A + A ^ *} {2} \ & =\ frac {A-A ^ *}{ 2} \ & =\ frac {1} {2} (A-H) + \ frac {1} {2} (H-A ^ *) \ EEA \ eeex $ Zhi $ \ beex \ Bea \ Sen {A-\ re a} & \ Leq \ frac {1} {2} \ Sen {A-H} + \ frac {1} {2} \ Sen {H-A ^ *} \ & =\ frac {1} {2} \ Sen {A-H} + \ frac {1} {2} \ Sen {A ^ *-H }\\\\\\sen {A-H} \ quad \ sex {\mbox {by (1 )}}. \ EEA \ eeex $
[Zhan Xiang matrix theory exercise reference] exercise 4.15