1. (fan-Hoffman ). set $ A \ In M_n $, note $ \ re a = (a + A ^ *)/2 $. then $ \ Bex \ lm_j (\ re a) \ Leq s_j (A), \ quad j = 1, \ cdots, N. \ EEx $
Proof: For $ x \ In \ BBC ^ N $, $ \ beex \ Bea x ^ * (\ re a) suitable for $ \ Sen {x} = 1 $) X & = x ^ * \ frac {A + A ^ *} {2} X \ & =\ frac {1} {2} (x ^ * AX + x ^ * A ^ * X) \\&=\ Re (x ^ * Ax) \ quad \ sex {z \ In \ BBC \ rA Z ^ * = \ bar z} \ & \ Leq | x ^ * ax |\\& = | \ SEF {ax, x} | \\& \ Leq \ Sen {ax }. \ EEA \ eeex $ is characterized by Courant-Fischer very small and extremely large, $ \ beex \ Bea \ lm_j (\ re) & =\ Max _ {s \ subset \ BBC ^ n \ atop \ dim S = J} \ min _ {x \ in S \ atop \ Sen {x} = 1} X ^ * (\ re) X \ & \ Leq \ Max _ {s \ subset \ BBC ^ n \ atop \ dim S = J} \ min _ {x \ in S \ atop \ Sen {x} = 1} \ Sen {ax }\\& = s_j (). \ EEA \ eeex $
[Zhan Xiang matrix theory exercise reference] exercise 4.1