[Zhan Xiang matrix theory exercise reference] exercise 4.10

10. set $ a, B \ In M_n $ and $ AB $ as the Hermite matrix, then, for any undo norm $ \ Bex \ Sen {AB} \ Leq \ Sen {\ re (BA )}. \ EEx $

 

 

 

Proof: (1 ). first prove $ \ Bex x \ prec Y \ rA | x | \ prec_w | Y |. \ EEx $ in fact, the $ x \ prec y $ knows $ \ beex \ Bea X & = Ay \ quad \ sex {: \ mbox {double random matrix }\\\&=\ sum \ al_kp ^ Ky \ quad \ sex {\ al_k \ geq 0, \ sum \ al_k = 1, \ P ^ k \ In \ pi_n, \ mbox {By Birkhoff theorem }\\\&=\ sum _ {\ Sigma \ In s_n} \ Al _ \ Sigma Y _ \ Sigma, \ EEA \ eeex $ and $ \ Bex | x | \ Leq \ sum _ {\ Sigma \ In s_n} \ Al _ \ Sigma | Y _ \ Sigma | = \ sum _ {\ Sigma \ In s_n} \ Al _ \ Sigma Q ^ \ Sigma | Y | \ prec | Y | \ quad \ sex {q ^ \ Sigma \ In \ pi_n }. \ EEx $ hence Theorem 3.9 (II), $ \ Bex | x | \ prec_w | Y |. \ EEx $(2 ). original Certificate question. the Theorem dominated by the fan shall only be verified for $ \ Bex S (AB) \ prec S (\ re (BA )). \ EEx $ in fact, $ \ beex \ Bea S (AB) & =| \ LM (AB) | \ quad \ sex {AB: \ mbox {Hermite matrix }, \ LM (AB) \ mbox {real vector }\\\& = | \ LM (BA) |\\& \ prec_w | \ LM (\ re (BA )) | \\& \ quad \ sex {\ LM (BA) = \ re \ LM (BA) \ prec \ LM (\ re (BA )), \ mbox {Chapter 3, question 1; further evidence} (1) }\\& \ prec_w S (\ re (BA )) \ quad \ sex {\ mbox {inference 4.11 }}. \ EEA \ eeex $

[Zhan Xiang matrix theory exercise reference] exercise 4.10