[Zhan Xiang matrix theory exercise reference] exercise 4.2

2. (Thompson ). if $ a, B \ In M_n $ is set, the matrix $ U exists, V \ In M_n $ Yes $ \ Bex | a + B | \ Leq u | A | u ^ * + v | B | V ^ *. \ EEx $

 

 

 

Proof: (1 ). the conclusion is valid only when $ C \ equiv A + B $ is positive. in fact, for the general $ C $, there is a $, a semi-Definite Matrix $ p $, make $ \ Bex c = WP \ rA P = w ^ * (A + B ). \ EEx $ and $ \ beex \ Bea | a + B | & = | c | \ & = p \ & \ Leq u | w ^ * A | u ^ * + v | w ^ * B | V ^ * \\& = u | A | u ^ * + v | B | V ^ *. \ EEA \ eeex $(2 ). when $ C $ is semi-positive, $ \ beex \ Bea | a + B | & = c = \ re c \ & = \ re a + \ re B \ & \ Leq u | A | u ^ * + v | B | V ^ * \ quad \ sex {\ mbox {by question 1st ,} S (A) = \ LM (| A |), \ mbox {and (3 )}}. \ EEA \ eeex $(3 ). conclusion: If $ X and Y $ are the Hermite arrays of $ N $, their feature values are $ \ Bex \ lm_j (x) \ Leq \ lm_j (y ), \ quad j = 1, \ cdots, N. \ EEx $ There Is A $, making $ \ Bex x \ Leq uyu ^ *. \ EEx $ in fact, there is a $ U_1, u_2 $, making $ \ Bex u_1xu_1 ^ * = \ diag (\ lm_1 (x), \ cdots, \ lm_n (x) \ Leq \ diag (\ lm_1 (Y), \ cdots, \ lm_n (y) = u_2yu_2 ^ *. \ EEx $ take $ u = U_1 ^ * u_2 $, then $ \ Bex x \ Leq uyu ^ *. \ EEx $

[Zhan Xiang matrix theory exercise reference] exercise 4.2