《矩阵理论—知识点详解》第四章 矩阵分解（4.1）矩阵的三角分解

1设P可逆,且‖P|<1,则‖Al引PA|l或 A|b=1APl均为自相容的矩阵范数 Proof:容易证明所定义的映射都是矩阵范数, 下面证明它们是相容的 ABlP4Bl= PAP PB llsP4.‖P.‖PB PA‖·‖PB|=‖A|l‖B| ‖AB|b=4BPll= PP BP llosA‖.‖P‖.‖BPl S‖AP‖‖BP|=‖41bB|b

返回 1 1 1 1. , || || 1, || || || || || || || || . Pr : , . || || || || || || || || . || || . || || || || . || || || || || || . a b a a a P P A PA A AP oof AB PAB PAP PB PA P PB PA PB A B −   − −      = = = =   = 设 可逆 且 则 或 均为自相容的矩阵范数 容易证明所定义的映射都是矩阵范数 下面证明它们是相容的 1 1 || || || || || || || || . || || . || || || || . || || || || || || . b b b AB ABP APP BP AP P BP AP BP A B − −     = =   =

2设4=A,则‖A|2SA|1=A≤nA|2 证明:由于A=A,所以‖A4=A‖ A2=r(AA=Amax(A"A)<A All1 圳A1‖|A4l1=4m2,故‖Al2sAⅢ

返回 2 1 2 , || || || || || || || || . H A A A A A n A =  =   2.设 则 2 2 max 1 2 1 1 1 2 1 || || ( ) ( ) || || || || || || || || , || || || || . H H H H A r A A A A A A A A A A A = =    =  故 1 1 : , || || || || || || H H 证明 由于 所以 A A A A A = = = 

A"=r(4A)=an(40≥ max max,nilai I2、max∑m142 2 2 ‖A 故A1≤nA2

返回 2 2 2 max 2 2 2 1 2 2 1 1 2 2 2 2 2 || || ( ) ( ) max | | || || | | max | | max ( | |) || || , || || || || . H H i n m i ij ij j n n i ij i ij j j A r A A A A n A a a n n n n a a n n A A n A n   = = =   = =  = =  =  =       故

41特征值界的估计 定理1(Shur不等式)设A∈C的特征值为 92 n,则 ∑|1≤Σ214;AF 且等号成立当且仅当为正规矩阵 证:A∈Cm以→A=UTUh ∑A42=t1t2+∑t2=m(rm) L≠J

返回 4.1 特征值界的估计 证： n n A C   H A = UTU  =  = = n i i i n i i t 1 2 1 2 |  | | |   +  =  n i j i j n i i i t t 2 1 2 | | | | tr(T T) H = 定理 1 (Shur不等式) 设AC nn 的特征值为 2 1 1 2 1 2 | | | | || ||F n i n j i j n i  i    a = A = = =  1 ,2 ,  ,n ,则 且等号成立当且仅当A为正规矩阵

A=UTUH→→AA=U(mHm)Uh tr(AA=tr(TT) ∑412≤mr(m7)=m(44)=4■

返回 H A = UTU H H H A A = U(T T)U  = n i i 1 2 |  | tr(T T) H  tr(A A) tr(T T) H H = tr(A A) H = 2 || || = A F

B=(4+A),C=(4-A") 2 2 A,B,C的特征值分别为41,A2,…,4n {A1,p2,…,n3,{iy1,iy2,…,iyn},且满足 1A1A2≥…2n, 1≥p2≥…2风n2, n1≥y2≥…≥yn

返回 1 1 ( ), ( ) 2 2 H H B A A C A A = + = − , , { , , , }, A B C的特征值分别为1 2  n {1 ,2 ,  ,n },{i 1 ,i 2 ,  ,i n }, 且满足 | | | | | |, 1  2  n 1 2 ,       n 1 2 . n      

定理2( Hirsch)设4∈C的特征值为A1, Dn i snmax aii b, 2) Reni s n max bii b 3)Imnisnmax ci b

返回 定理 2 (Hirsch) 设AC nn 的特征值为 2 ,  ,n ,则 , 1 1)| | max | |, , i j i j i  n a 2)| Re | max | |, , i j i j i  n b 3)|Im | max | |, , i j i j i   n c

证 1)|42s24∑2|a12n2mx|an2 Mniksnmax ai l 2)A∈C"→UHAU=T, UHAHU=TH U BU=U (A+ AU=-(T+r) U CU 01(4-1=- 2

返回 证： n n A C  2)  H H H H U AU = T, U A U = T ( ) 2 1 ( ) 2 1 U BU U A A U T T H H H H = + = + ( ) 2 1 ( ) 2 H 1 H H H U CU = U A− A U = T −T      = n i i i 1 2 2 1) |  | |  |    = = n i n j aij 1 1 2 | | 2 , 2 max | | ij i j  n a | | max | | , i j i j i  n a

2+x ∑|b 2 j=1 A:-见 2 =li=12 lj=l ∑!Re412s∑hn2mx|b12 ∑|mx2≤∑∑cn2≤n2max|cn2

返回                = = = = = = n i n j i j n i i n i n j i j n i i c b 1 1 2 1 2 1 1 2 1 2 |Im | | | | Re | | |   2 , 2 max | | ij i j  n b 2 , 2 max | | ij i j  n c         +   =   −  +   =   + = = = − = = = = = − = = n i n j i j n j j i i j n i i i n i n j i j n j j i i j n i i i c t b t 1 1 2 1 1 1 2 1 2 1 1 2 1 1 1 2 1 2 | | 2 | | | 2 | | | 2 | | | 2 |    

2) Rehi is nmax bi l 3)| i ks nmax c 定理3( Bendixson)设A∈R,则4的任一特 值满足 I Im a: k/n(n-1) max

返回 3)|Im | max | | , i j i j i   n c 2)| Re | max | | , i j i j i  n b    定理 3 (Bendixson) 设A R nn ,则A的任一特 值i 满足 max | | 2 ( 1) |Im | , i j i j i c n n −  