矩阵论3_λ矩阵与矩阵的Jordan标准型.pdf

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1、1nÙλÝÝJordanIO/3.1õª•½Â3.1.1P´ê§n´Kê§λ´©i§/ªLªnn−1f(λ)=anλ+an−1λ+···+a1λ+a0Ù¥ai∈P(i=0,1,···,n)§¡êPþõª§êPþõªNPP[λ]¶n¡f(λ)gê§P∂(f(λ))=n;ean=1§K¡f(λ)Äõª"•õª!!~!¦!ê¦9Ù$5Æ•½n3.1.1f(λ),g(λ)∈P[λ]g(λ)6=0§K3õªq(λ),r(λ)∈P[λ]¦f(λ)=q(λ)g(λ)+r(λ)Ù¥r(λ)=0½r(λ)6=0∂(

2、r(λ))<∂(g(λ))¶q(λ)¡g(λ)Øf(λ)û§r(λ)¡g(λ)Øf(λ){ª"•½Â3.1.2f(λ),g(λ)∈P[λ]XJ3h(λ)∈P[λ]¦f(λ)=h(λ)g(λ)K¡õªg(λ)Øf(λ)§Pg(λ)

3、f(λ)¶g(λ)¡f(λ)Ϫ¶f(λ)¡g(λ)ª"XJg(λ)ØUØf(λ)§Pg(λ)-f(λ)•5¿«OõªØêØ"Xµ3P[λ]¥§43

4、4,4=×33
573

5、5λ+7,5λ+7=λ+×333
3

6、8λ2+λ+2,8λ2+λ+2=8λ2+1λ+2×3333AO

7、/§g

8、f(λ),∀"g∈P,∀f(λ)∈P[λ].•5XJg(λ)

9、fi(λ)(i=1,···,m)§Kg(λ)

10、(u1(λ)f1(λ)+···+um(λ)fm(λ))§Ù¥ui(λ)∈P[λ](i=1,···,m)•úϪ§úϪ§p§úª§úª3.2λÝ9Ù3-eIO/3.2.1λÝÄVg•½Â3.2.1aij(λ)(i=1,2,···,m;j=1,2,···,n)´êPþõª§±aij(λ)m×nÝa11(λ)a12(λ)···a1n(λ)a21(λ)a22(λ)···a2n(λ)A(λ)=

11、.........am1(λ)am2(λ)···amn(λ)¡õªÝ½λݧõªaij(λ)(i=1,2,···,m;j=1,2,···,n)¥pgê¡A(λ)gê§êPþm×nλÝNPP[λ]m×n•êiÝ´λÝA~§A∈Cn×nAÝλI−A´1gλÝ•λÝ!{!~{!¦{!ê¦9Ù$5K•XJm×nλÝA(λ)gêk§KA(λ)L«kk−1A(λ)=Akλ+Ak−1λ+···+A1λ+A0Ù¥Ai(i=0,1,···,k)´m×nêiݧ¿Ak6=0¶~X−λ+1λ201−10102A(λ

12、)==λ+λ+λλ001100•λÝ1ª!fª!ê{fª$Ú5êiÝaq"fª½ÂµA(λ)∈P[λ]m×n§3A(λ)¥Àr1r(1≤r≤n)§dù1Ú?U5'¤r1ª¡A(λ)rfª"•'Xµλ+1λ2+1λ23λ−13λ2−1λ2+2λk32fªλ+1λ2+1=4λ2−4λ3λ−13λ2−1λ+1λ2=−2λ3+4λ2+2λ3λ−1λ2+2λλ2+1λ2=−2λ4+2λ3+2λ2+2λ3λ2−1λ2+2λ•½Â3.2.2A(λ)∈P[λ]m×n§XJA(λ)¥kr(1≤r≤min{m,

13、n})fªØ"§¤kr+1fª£XJk{¤"§K¡A(λ)r§Prank(A(λ))=r•~A∈Cn×n§

14、λI−A

15、´λngÄ1õª§Ïd

16、λI−A

17、´λn§o´÷"•½Â3.2.3A(λ)∈P[λ]n×n§XJ3nÝB(λ)¦A(λ)B(λ)=B(λ)A(λ)=IK¡λÝA(λ)´_§¿¡B(λ)A(λ)_ݧPA(λ)−1•5¿µþã½Â=3P[λ]n×n¥_"'Xµ−11−1λ+11=λ−1λ−121−2λ+1λ−1λ−13λkn©ªÝ¿Âe´_§3P[λ]2×2¥Ø´_"−1

18、λ+11−11=λ+21λ+2−λ−13P[λ]2×2¥´_"•½n3.2.1A(λ)∈P[λ]n×n§KA(λ)_¿©7^´

19、A(λ)

20、"~êy²75"A(λ)_⇓A(λ)B(λ)=B(λ)A(λ)=I⇓

21、A(λ)B(λ)

22、=

23、A(λ)

24、

25、B(λ)

26、=1⇓

27、A(λ)

28、´"~ê¿©5"

29、A(λ)

30、=d´"~ê⇓EA(λ)ÝA(λ)∗§KA(λ)1A(λ)∗=1A(λ)∗A(λ)=Idd⇓A(λ)−1=1A(λ)∗d3.2.2λÝÐC-•½Â3.2.4en«C¡λÝÐC£1¤λÝü1£¤p£2¤λÝ,1£

31、¤¦±"~êk£3¤λÝ,1£¤ϕ(λ),1£¤§Ù¥ϕ(λ)´õª•PÒ[i,j]L«1i,j1£¤pPÒ[i(k)]L«^"~êk¦1i1£¤PÒ[i+j(ϕ)]L«ò1j1£¤ϕ(λ)1i1£¤•n