矩阵乘积AB和BA的相关性质80
上亿文档资料，等你来发现
矩阵乘积AB和BA的相关性质80
第24卷第2期长春工业大学学报Vol124No.；矩阵乘积AB和BA的相关性质；王秀玉1,姜兴武2,曲波1；(1.长春工业大学基础科学学院,吉林长春1300；摘要:讨论了矩阵乘积AB与BA的特征值、特征向量；定义1设A∈Cm×n,B∈Cn×m,不妨设m；≤n；零解ΖA；-μ1Im；-μ2B；=0Ζ-μ10；-In；-μ1Im；=0Ζ；-1；-μB2In+μ1
　第24卷　第2期　　　　　　　　　　长　春　工　业　大　学　学　报　　　　　　　　　Vol124　No.2　　　2003年6月　　　JOURNALOFCHANGCHUNUNIVERSITYOFTECHNOLOGY　　　　　　　　　Jun12003文章编号:03)　　①矩阵乘积AB和BA的相关性质王秀玉1,　姜兴武2,　曲　波1(1.长春工业大学基础科学学院,吉林长春　.吉林商业高等专科学校,吉林长春　130062)摘　要:讨论了矩阵乘积AB与BA的特征值、特征向量及秩等的关系,并得到了矩阵的奇异值分解。关键词:特征值;特征向量;奇异值中图分类号:O151.2　　　文献标识码:A　　定义1　设A∈Cm×n,B∈Cn×m,不妨设mn≤n。对μ1,μ2∈C,若存在非零向量ξ∈C,η∈ξ=μ1η,Bη=μξ,则称(μC使得A21,μ2)为矩阵对(A,B)的一个特征值对,(ξ,η)A,于(μ1,μ2)=若AB1μ2,ξ=μBA121μ2AB与BA的特征值,η为AB属于μ1μ2的特征向量,ξ为BA属于μ1μ2的特征向量。m零解ΖA-μ1ImB-μ2B=0Ζ-μ10-In-μ1Im=0Ζ-1-μB2In+μ1BAμ　　|μ12In-BA|=0。ξ引理3若μ0是AB的特征值,BA1μ2≠i=μξ1μ2i,i=1,2,…,s,而且ξ1,ξ2,…,ξs线性无-1ξ关,记ηi=μ1Ai,i=1,2,…,s,则η1,η2,…,ηs也线性无关。-1ηξξ证明Bi=μ1BAi=μ2i,i=1,2,…,s,若k1η1+k2η2+…+kηss=0,两边用B作用得:ηηηξk1B1+k2B2+…+ksBs=0,也就是:μ2k11+1AB与BA的特征值与特征向量的关系mn-In-BA|=λ引理1f(λ)=|λ)=|λg(λIm-AB|。g(λ),证明AλIm-ABIn(-当λ≠0时,有:=(-1)-mn+n(n+1)AλImInB=μξ0,则k1=k2=…=2k22+…+μ2kξss=0,μ2≠ks=0。01) mn|λIm-AB|=AλImInB|λImλImBAB|,又mn+m(m+1)=ηξ在引理3中,ABi=μ2Ai=μ1μ2ηi,i=1,2,…,s,由引理1和引理3可得。定理1设A∈Cm×n,B∈Cn×m,则λ≠0作为AB与BA的特征值,有相同的代数重数,同时也有相同的几何重数。-1In-λBA=(-1)mλIn-λ-1BA=n-=λmm-n(-1)mnλIn-BA|λIn-BA|,从而有:λ|λImm)=-AB|,当λ=0时,若m&n,由于|BA|=0,亦有f(λn-mn-λ),若m=n,则有|BA|=|AB|,也有f(λ)=λg(λ2　AB与BA的秩与最小多项式引理4μ0,(μ1≠1,0)为(A,B)的特征值对ΖdimN(A)dimN(BA)Ζrank(BA&rank(A)。ξ=μξ=0ABA1η证明有非零解Ζ有非ξμηA=η1B=0零解ΖN(A)≠N(BA)ΖdimN(A)&dimN(BA)Ζrank(BA)&rank(A)。)。g(λ若μ0,(μ1μ2≠1,μ2)为矩阵对(A,B)的特征值对Ζμ1μ2为BA的特征值。引理2证明ξ=μηA1η=μξB2有非零解η=μμηAB12η=μξB2有非①收稿日期:作者简介:王秀玉(1965-),女,吉林蛟河人,长春工业大学副教授,理学硕士,主要从事矩阵理论研究.第2期　　　　　　　　　　　　　王秀玉等:矩阵乘积AB和BA的相关性质65引理5μ0,(0,μ2≠2)为(A,B)的特征值对Ζrank(AB)rank(B)。ξ=0η=0AAB证明有非零解Ζ有非η=μ2ξη=μξBB2零解ΖN(AB)≠N(B)ΖdimN(B)&dimN(AB)Ζrank(AB)&rank(B)。记rank(A)=rA,rank(B)=rB,rank(AB)=rAB,rank(BA)=rBA,则有:N(A)&N(BA),N(B)证明设BA的最小多项式为:ψ(λ)=lλ),由于φ(λ)为AB的最小多项式,故有p(λφ(AB)=0,即(AB)k[(AB)t+qt-1(AB)t-1+…+q1(AB)+q0I]=0,左乘B,右乘A得:B(AB)[(AB)+qt-1(AB)tt-1k+…+q1(AB)+q0I]A=[(BA)+qt-1(BA)tt-10,即(BA)k[(BA)t+1+qt-1(BA)t+…+q1(BA)2k+1+q0(BA)]=0,(BA)+&N(AB)。引理6rA+rB≤m+rBA。证明　当rBA=rA时,有rA+rB=rBA+rB≤m+rBA,当rBA&rA时,取N(A)的一组基为:ζ1,ζζ2,…,ζn-rA,把它扩充成N(BA)的一组基为:1,ζ2,…,ζn-rA…+q1(BA)+q0I]=0。)整除λφ(λ),而　　由最小多项式的性质得ψ(λλ不能整除q(λ),所以必有p(λ)整除q(λ),反)整除p(λ),因而有p(λ)=q过来同理可证q(λl(λ),即得ψ(λ)=λ)。q(λ推论　AB与BA的Smithλ,,BA以λ≠0为块均相同。,ζn-rA+1,…,ζn-rBA,取μ0,μ1≠1∈-1ζC,(μ1,0)为(A,B)的特征值对,记βi=μ1Ai,i=1,2,…,n-rBA,则β1=β2==n-rA=,βn-rA+1,+2,,rA+2rA+1rBA关rBA×nmn设A∈Cm,μ∈C,υ∈C,σ≥0rk1βn-rA+2n-+krA-βn-rBA=0,也　　定义2-1ζ就是k1μ1An--1ζ+k2μ1An-rA+1rA+2+…+rA+2rBAkrA-rBA-1μζ1An-rBArBA=0,k1ζn-+k2ζn-+…=0,ζ+krA-rBAn-故βn-rA+1∈N(A),k1=k2=…=krA-rA+2,βn-,…,βn-rBAβ线性无关。又Bi=0,i=n-rA+1,…,n-rBA,故有n-rBA-(n-rA)≤m-rB,rA+rB≤m+rBA。υ=σμ,A3μ=συ,则称σ为A的奇异值,μ和使Aυ分别称为A的左、右奇异向量。由引理2知,σ≠0为A的奇异值,当且仅当2σ为AA3的特征值,且A的关于σ的线性无关的左奇异向量的个数与A的关于σ的线性无关的右奇异向量的个数相同。υ=σμA3υAμ=σ引理7证明rA+rB≤n+rAB。Ζ32AAμ=σμ3υAμ=σ,因此有引理8。当rAB=rB时,rA+rB=rA+rAB≤n+rAB,当rAB&rB时,取μ0,(0,μ2≠2)为(A,B)的特征值对,取N(B)的一组基为:η1,η2,…,ηm-rB,把它扩充成N(AB)的一组基为:η1,η2,…,ηm-γm-rB引理8σ≠0为A的奇异值,AA3μi=2σμi,i=1,2,…,s,且有(μi,μj)=1　i=j0　i≠j-13,υ,s,i=σAμi,i=1,2,…,ηm-rB+1,…,ηm-rAB,取γi=μ2-1Bηi,rB+1i,则(υi,υj)==m-rB+1,m-rB+2,…,m-rAB,则γm-rB+21　i=j0　i≠j,…,γm-rABγi=0,i=m-线性无关,又A证明rB+1,m-rB+2,…,m-rAB,从而有m-rAB-(m-rB)≤n-rA,即rA+rB≤n+rAB。-13-13(υAμAμi,υj)=(σi,σj)=σ-2μj3AA3μi=μj3μi=(μi,μj)。2设AA3的非零且互不相同的特征值为σ1&2232σμ2&…&σs&0,AAμij=σiij,j=1,2,…,ti,i=由引理6及引理7可得。定理2定理3设A∈Cm×n,B∈Cn×m,则有rA+rB≤min{m+rBA,n+rAB}。设A∈Cm×n,B∈Cn×m,Ind(AB)=)k,Ind(BA)=l,矩阵AB的最小多项式为:φ(λkktt-1)=λ(λ=λq(λ+qt-1λ+…q1λ+q0),q0≠l)=λ)。0,则BA的最小多项式为:ψ(λq(λ1,2,…,s,且(μij,μik)=1　j=k0　j≠ks,i=1∑ti=r,则有引理9。　　引理9　μ11,μ12,…,μ1t1,μ21,μ22,…,μ2t2,…,μs1,μs2,…,μsts为标准正交向量组。66长　春　工　业　大　学　学　报　　　　　　　　　　　　　　　第24卷22333σμμij=μpkAAμi(μij,μpk)=σipkij=证明(AA3μpk)32μij=(μij,AA3μpk)=(μij,σμpk)=pυr+1,υr+2,…,υn,则υ11,υ12,…,υ1t1,…,υs1,υs2,…,υsts,υr+1,υr+2,…,υn为标准正交向量组,记V=(υ11,υ12,…,υ1t1,…,υs1,υs2,…,υsts,υr+1,υr+2,2σp(μij,μpk),当i≠p时,(μij,μpk)=0。取N(A3)=N(AA3)的标准正交基为:μr+1,μr+2,…,μn,则μ11,μ12,…,μ1t1,μ21,μ22,…,μ2t2,…,μs1,μs2,…,μsts,μr+1,μr+2,…,μn为标准正交向量组,记U=(μ11,μ12,…,μ1t1,…,μs1,μs2,…,μsts,μr+1,μr+2,…,μn)为酉矩阵,记-13υij=σiAμij,j=1,2,…,ti,i=1,2,…,s,则…,υn),则V为酉矩阵,则有定理4。∑0定理4U3AV=,∑=diag(σ1,σ2,0…,σr)。参考文献:[1]　程云鹏.矩阵论[M].西安:西北工业大学出版υ11,υ12,…,υ1t1,…,υs1,υs2,…,υsts为标准正交向量组,取N(A)=N(A3A)的一组标准正交基为社,20001SomePropertiesoftheProducts212　QUBo1(1.Basic,UniversityofTechnology,Changchun130012,C2.JilinCommercialCollege,Changchun130062,China)Abstract:Therelationshipbetweentheeigenvalue,eigenvectorandtherankoftheproductofmatricesABandBAisdiscussed,andthefactorizationofthesingularvaluesofmatrixAisobtained.Keywords:singularvalue.包含各类专业文献、行业资料、中学教育、应用写作文书、高等教育、文学作品欣赏、幼儿教育、小学教育、矩阵乘积AB和BA的相关性质80等内容。
　了矩阵 A、B 在什么条件下其特征值的乘积是 AB 与 BA 的特征值, 以及矩阵...给出 了矩阵 AB 与 BA 相似的情况下的一些相关性质的讨论与推广,我相信这些... 　Hadamard乘积矩阵的一些性... 5页 5财富值 矩阵乘积AB和BA的相关性质 3页 1...? 2.收敛的无穷乘积的性质 2.1 收敛的无穷乘积的性质定理 2 定理 3 [ 4]... 　回顾导入(20 分钟)――复习矩阵的运算求乘积 AB 与 BA 及 AI 与 IA 引入...二、主要教学过程(60 分钟,其中学生练习 20 分钟)三:矩阵乘法的性质: 1) ... 　比如乘法 AB 一、1)用 A 的第 1 行各个数与 B 的第 1 列各个数对应...(A)是环中的可逆元那么公式(*)表明 性质对 n×n 的矩阵 A 和 B,有: 1... 　Hadamard 乘积及 Kronecker 乘积的定义,证明 了它们的一些运算性质,并介绍了它们...2)交换律 矩阵的乘法不适合交换律,即一般说来, AB ? BA 。 这是由于,首先... 　,计算 AB,BA;(AB)C.A(BC ?0 2 ? ? ? ? ?2 3 ? ? ? (2)已知 ...第10课时矩阵乘法的的性... 3页 免费 2.线性变换的基本性质与... 暂无评价... 　? 定理 2.1.1 设 A、B 为 n 阶循环矩阵,则有: (1)乘积 AB 仍是循环矩阵,且满足乘法交换律,即 AB ? BA ;(2)若 A 可逆,则 A 的逆矩阵也是循环... 　4、矩阵乘法的定义和性质(1)当矩阵 A 的列数和 B 的行数相等时,则 A 和...② 矩阵乘法无交换律. 即 AB ? BA 乘法公式 一般地,由于交换性的障碍,小... 　a0 I . 性质 1.1 证明 设 两个循环矩阵 A, B 的乘积仍为循环矩阵,且 AB ? BA . A ? a0 I +a1D ? a2 D 2 ? ?? ? an ?1D n ?1 ? f (...In-GameDota 2 - Join
Shootingstar.Weiser.ワイス
KARL GAMING !!!
Last Online 2 mins ago
вυ?αвσ?? ?
Last Online 19 mins ago
Cock Processor
Last Online 1 hrs, 4 mins ago
ShomariBon
Last Online 1 hrs, 10 mins ago
Last Online 1 hrs, 40 mins ago
Last Online 1 hrs, 42 mins ago
J O N G ッ
Last Online 1 hrs, 46 mins ago
_Шусстрый_:D
Last Online 1 hrs, 55 mins ago
Trauma The Crow
Last Online 2 hrs, 11 mins ago
Last Online 2 hrs, 49 mins ago
Capitan Butt-Head
Last Online 3 hrs, 5 mins ago
Last Online 3 hrs, 32 mins ago
Last Online 3 hrs, 37 mins ago
±|BalanceBreakeR|±
Last Online 5 hrs, 5 mins ago
Galawang Sebastian
Last Online 5 hrs, 17 mins ago
&font face=&ravie&&Mr.
Last Online 5 hrs, 31 mins ago
m444rteeens
Last Online 5 hrs, 32 mins ago
Last Online 5 hrs, 39 mins ago
Last Online 5 hrs, 54 mins ago
Galawang Brezzy ?
Last Online 6 hrs, 7 mins ago
Last Online 6 hrs, 24 mins ago
&i&&b&&u&MAMBA
Last Online 6 hrs, 49 mins ago
? ASDFGHJKL ?
Last Online 6 hrs, 52 mins ago
Last Online 7 hrs, 2 mins ago
Last Online 7 hrs, 3 mins ago
Last Online 7 hrs, 6 mins ago
Last Online 7 hrs, 9 mins ago
Last Online 7 hrs, 12 mins ago
Indominus Rex
Last Online 7 hrs, 29 mins ago
My eyebrow is growing back
Last Online 7 hrs, 33 mins ago
Last Online 7 hrs, 36 mins ago
Ckl Pe?aflorida Serna
Last Online 7 hrs, 42 mins ago
N7 Hannah montana?
Last Online 7 hrs, 53 mins ago
'*' staff of Sanity'';'
Last Online 7 hrs, 56 mins ago
PrettyGirlTuthsiツ
Last Online 7 hrs, 58 mins ago
Jagiella Dhayne!
Last Online 7 hrs, 59 mins ago
silent please
Last Online 8 hrs, 2 mins ago
ceasarGwapz &3
Last Online 8 hrs, 3 mins ago
Last Online 8 hrs, 6 mins ago
DAYUMASIG LOW PRIO SHIT
Last Online 8 hrs, 15 mins ago
Maynard Vega Villanueva
Last Online 8 hrs, 16 mins ago
Last Online 8 hrs, 17 mins ago
laurenceqwe
Last Online 8 hrs, 25 mins ago
theworldof NAS
Last Online 8 hrs, 28 mins ago
Last Online 8 hrs, 28 mins ago
Last Online 8 hrs, 33 mins ago
Last Online 8 hrs, 35 mins ago
SteadyPompipi
Last Online 8 hrs, 43 mins ago
G[*]D[A]MN_I+
Last Online 8 hrs, 46 mins ago
[KWIC] |(TM)JARED O. DEMETRIAL(TM)|
Last Online 8 hrs, 49 mins ago
ThE mEchAniC !
Last Online 8 hrs, 52 mins ago
P a r k - Yeh - Eun &3
Last Online 9 hrs, 2 mins ago
??????? ??
Last Online 9 hrs, 7 mins ago
The Game Is Over
Last Online 9 hrs, 14 mins ago
&u&Ms.Maegan
Last Online 9 hrs, 17 mins ago
Last Online 9 hrs, 22 mins ago
&font face='impact'&&u& xGreEd
Last Online 9 hrs, 23 mins ago
MONSTER.Xnn[A]
Last Online 9 hrs, 26 mins ago
Last Online 9 hrs, 30 mins ago
Last Online 9 hrs, 42 mins ago
&::::::::::::{8}
Last Online 9 hrs, 43 mins ago
ツ ???? ツ
Last Online 9 hrs, 43 mins ago
Last Online 9 hrs, 50 mins ago
Last Online 10 hrs, 14 mins ago
Last Online 10 hrs, 17 mins ago
Last Online 10 hrs, 20 mins ago
SkyPink ./.
Last Online 10 hrs, 22 mins ago
-GodExecuter-
Last Online 10 hrs, 25 mins ago
Last Online 10 hrs, 35 mins ago
[POGI]- 3 + 1 = 4
Last Online 10 hrs, 41 mins ago
Last Online 10 hrs, 51 mins ago
Last Online 10 hrs, 51 mins ago
Last Online 10 hrs, 57 mins ago
Last Online 11 hrs, 1 mins ago
Last Online 11 hrs, 7 mins ago
*Extreme-&mahou/Jao
Last Online 11 hrs, 8 mins ago
EMP-BosS ツ
Last Online 11 hrs, 19 mins ago
? iD-ム-R-O-K ツ
Last Online 11 hrs, 47 mins ago
&font face='GOUDY STOUT'& LOSER
Last Online 11 hrs, 47 mins ago
Last Online 12 hrs, 4 mins ago
[Mighty ******* And Carry].[M.S.
Last Online 12 hrs, 10 mins ago
loading.....
Last Online 12 hrs, 16 mins ago
Last Online 12 hrs, 17 mins ago
&i&&u&GaLaWanG,-.JeaN&i&&u&
Last Online 12 hrs, 18 mins ago
*_*Jarhyl 4EVER*_*
Last Online 12 hrs, 41 mins ago
-Revenge Of Mc
Last Online 12 hrs, 50 mins ago
SON's OF SATANAS ! ./.
Last Online 13 hrs, 10 mins ago
Last Online 13 hrs, 22 mins ago
-Snox_QWE-
Last Online 13 hrs, 23 mins ago
Chico Dragon
Last Online 13 hrs, 32 mins ago
You're in Deep Shit
Last Online 13 hrs, 45 mins ago
I LoVe CeBu
Last Online 14 hrs, 35 mins ago
Last Online 15 hrs, 9 mins ago
Last Online 15 hrs, 20 mins ago
oNieR_gWApO
Last Online 15 hrs, 26 mins ago
&Im Newbie&
Last Online 16 hrs, 8 mins ago
CHUPAHPING-乂
Last Online 16 hrs, 28 mins ago
Nina CoLLiamat &3
Last Online 16 hrs, 32 mins ago
Post No Bill
Last Online 17 hrs, 7 mins ago
S h i r a y u k i
Last Online 17 hrs, 17 mins ago
Last Online 17 hrs, 19 mins ago
Last Online 17 hrs, 44 mins ago
&i&&u&& Pop Danthology &
Last Online 17 hrs, 53 mins ago
Man with a 1k Moves
Last Online 18 hrs, 48 mins ago
Last Online 19 hrs, 17 mins ago
Last Online 20 hrs, 25 mins ago
Last Online 21 hrs, 27 mins ago
! WHo CaReSZ !
Last Online 21 hrs, 29 mins ago
&font face=&magneto&&JaH_LaZaDa&
Last Online 21 hrs, 33 mins ago
Last Online 21 hrs, 43 mins ago
Last Online 22 hrs, 3 mins ago
Last Online 22 hrs, 18 mins ago
Glenford Kun
Last Online 22 hrs, 24 mins ago
Ninja bakukang
Last Online 24 hrs, 41 mins ago
NATHAN.DanWin :D
Last Online 26 hrs, 48 mins ago
Chocodo(TM)[bOgz]
Last Online 30 hrs, 9 mins ago
4K SCRUBS !
Last Online 30 hrs, 55 mins ago
Last Online 31 hrs, 15 mins ago
FraFries ツ
Last Online 31 hrs, 25 mins ago
Anygma (&&) :) ...../.....
Last Online 33 hrs, 9 mins ago
Last Online 33 hrs, 29 mins ago
Supp Nigga ,,/,,
Last Online 33 hrs, 32 mins ago
&font face=&Ravie&&BooChooii
Last Online 33 hrs, 46 mins ago
heterochromia
Last Online 34 hrs, 43 mins ago
GraceNel &3
Last Online 34 hrs, 48 mins ago
Liza Soberano&3
Last Online 34 hrs, 53 mins ago
Last Online 35 hrs, 6 mins ago
#04-[Janine]-
Last Online 35 hrs, 7 mins ago
Last Online 35 hrs, 7 mins ago
[X] Rekt [ ]Not rekt #swag
Last Online 35 hrs, 16 mins ago
Standin.Arib
Last Online 35 hrs, 34 mins ago
Standin.Asdasd
Last Online 36 hrs, 3 mins ago
Last Online 36 hrs, 53 mins ago
█ █\/█ Carl.&3
Last Online 38 hrs, 24 mins ago
?ッY??O???L???O?ッ
Last Online 38 hrs, 56 mins ago
Chippy.Gigo
Last Online 38 hrs, 57 mins ago
Mski.Net|Zaki102.Razer
Last Online 39 hrs, 6 mins ago
Last Online 39 hrs, 7 mins ago
Twerk It Like Shit!!
Last Online 40 hrs, 5 mins ago
,.o'?)Ｈｏｏｋツ
Last Online 40 hrs, 7 mins ago
s4.Sebastian
Last Online 40 hrs, 19 mins ago
VALV E(TM)
Last Online 41 hrs, 32 mins ago
Pagina Venis
Last Online 43 hrs, 54 mins ago
Last Online 44 hrs, 8 mins ago
--=]Fr[A]G [E]XeCuteR o.O :P [=-
Last Online 44 hrs, 25 mins ago
muere maldito insecto
Last Online 44 hrs, 29 mins ago
?GaLaWa∩G?VenZam.
Last Online 45 hrs, 14 mins ago
-smile-k-din-
Last Online 2 days ago
Last Online 2 days ago
Jason Derulo
Last Online 2 days ago
The Answer
Last Online 3 days ago
NoobFromUA
Last Online 3 days ago
B I N G G O
Last Online 3 days ago
Last Online 3 days ago
fuck this mmr !!!
Last Online 3 days ago
Last Online 3 days ago
Master*Pred
Last Online 3 days ago
Last Online 3 days ago
¤ JoKeRMaN ¤
Last Online 3 days ago
Led?hkarl.
Last Online 3 days ago
TryToHitMe NooB
Last Online 3 days ago
Last Online 4 days ago
Last Online 4 days ago
l_Kryptonite_l
Last Online 4 days ago
NoobFromPH
Last Online 4 days ago
XProximity
Last Online 4 days ago
asaka!!!!!!
Last Online 4 days ago
2/M Zamora,Ryan
Last Online 4 days ago
Last Online 4 days ago
FantaSy iS THE ReaLLitY
Last Online 4 days ago
QUIT DOTA &/3
Last Online 4 days ago
{*sky.ark*}
Last Online 5 days ago
Last Online 5 days ago
Last Online 5 days ago
Last Online 5 days ago
UNKWN.JR11 ツ
Last Online 5 days ago
Last Online 5 days ago
???iKa?mar???
Last Online 5 days ago
Last Online 5 days ago
Can i give you a try? :&?
Last Online 6 days ago
Galawang Pogi
Last Online 6 days ago
&:わ???レiN?:&
Last Online 6 days ago
Last Online 6 days ago
√??Ωεss<>√??Ωεss√??
Last Online 6 days ago
SNSD.TIFFANY HWANG ?
Last Online 6 days ago
Last Online 8 days ago
Hii Mak Mak :3
Last Online 8 days ago
Last Online 9 days ago
Last Online 9 days ago
Last Online 10 days ago
Last Online 10 days ago
{GM}LuCkY.NalNel
Last Online 10 days ago
One_Hit _Delete*
Last Online 11 days ago
Last Online 11 days ago
WalangForever
Last Online 11 days ago
Zhang Yixing
Last Online 14 days ago
BarKa.CuterzZz.KnocKii
Last Online 14 days ago
Last Online 15 days ago
Last Online 15 days ago
*Never_Give_Up*
Last Online 16 days ago
[Chocodo(TM)[Brother]
Last Online 17 days ago
Last Online 17 days ago
KingSman_)
Last Online 18 days ago
Standing.Chammez
Last Online 18 days ago
Last Online 18 days ago
♀ Fly High ♂
Last Online 18 days ago
Last Online 19 days ago
C[L]oud Gaming xD
Last Online 20 days ago
322.Jah^420^man
Last Online 21 days ago
Last Online 22 days ago
? X-perto ?
Last Online 23 days ago
Last Online 23 days ago
Last Online 23 days ago
Last Online 25 days ago
~neOn_bLacK~
Last Online 25 days ago
JustDoiT???????
Last Online 26 days ago
Last Online 27 days ago
Last Online 27 days ago
Last Online 30 days ago
PanterBeLL
Last Online 31 days ago
-PewDiePie-
Last Online 32 days ago
&_& RAMPAGE &_&
Last Online 32 days ago
Last Online 32 days ago
aleeeeeeexis
Last Online 33 days ago
Last Online 33 days ago
Mr.GaMbLeR
Last Online 34 days ago
Im so Noob hehe XD !!!!
Last Online 36 days ago
Last Online 37 days ago
Last Online 38 days ago
Last Online 39 days ago
Last Online 40 days ago
Last Online 41 days ago
Amazing.Chupappey &3
Last Online 42 days ago
Last Online 45 days ago
Last Online 47 days ago
Last Online 47 days ago
&HEY SHIT&
Last Online 48 days ago
EL FELEBUSTISIALMO
Last Online 50 days ago
......IM FUCKing bitch...
Last Online 50 days ago
The Last Skill
Last Online 52 days ago
QZ_AUBREY&3
Last Online 60 days ago
Last Online 62 days ago
Last Online 63 days ago
Last Online 65 days ago
Na?Vi Team
Last Online 66 days ago
Last Online 71 days ago
KentJoy???
Last Online 72 days ago
?GALAWA∩G?KEUR?Tツ
Last Online 72 days ago
Last Online 74 days ago
EaT.My.ShiT
Last Online 75 days ago
&i& StandiN SumaiL &u& NeeDs Me
Last Online 77 days ago
CHuPa TUNLON
Last Online 83 days ago
Last Online 84 days ago
Last Online 87 days ago
Gary Duck Face
Last Online 90 days ago
XXDaRvYxX56
Last Online 91 days ago
Last Online 97 days ago
[OS]Ekha Ageng Crieshantony
Last Online 99 days ago
Claire got this?
Last Online 104 days ago
ALWAYSWANNAFLY
Last Online 105 days ago
Last Online 107 days ago
WALKING_DEAD
Last Online 110 days ago
Last Online 113 days ago
-MG- ㄨ MasteR
Last Online 113 days ago
Last Online 117 days ago
Thewses James Matsanity
Last Online 125 days ago
`SpHinX``-Justin``
Last Online 126 days ago
Last Online 133 days ago
Last Online 133 days ago
Zzzzzzz[AFK]
Last Online 136 days ago
-MG - You Cant See Me
Last Online 136 days ago
expenser_virus
Last Online 142 days ago
Last Online 143 days ago
Last Online 143 days ago
KYle Idol :)
Last Online 143 days ago
?GALAWA∩G?RabZZ ツ
Last Online 144 days ago
SORY ! ~ !
Last Online 146 days ago
?? ?? ?? ?? R
Last Online 147 days ago
Christopher Chan
Last Online 151 days ago
WoO..DoTa !
Last Online 156 days ago
J o x h u [A]
Last Online 160 days ago
+AE★KEUR?T★ae+
Last Online 183 days ago
[RB] ' roccat
Last Online
Last Online
Ms.batiLLer
Last Online
Last Online
Last Online
Last Online
Last Online
Last Online
Last Online
Last Online
BurNing :3
Last Online
Im So Noob Shert's
Last Online
(GALS?)zAchEurry
Last Online
HL.SusuKimcil
Last Online
PointClickKiLL
Last Online
Last Online
Oggy&andTheCockroaches
Last Online
Eulz.Gaming
Last Online
Last Online
senzu is back
Last Online
Last Online
Rhea marie
Last Online
leave me alone
Last Online
VTS.Shorty
Last Online
Memetastic
Last Online}