Аналитическая геометрия и линейная алгебра. Умнов А.Е. - 205 стр.

UptoLike

ВУЗ: 

МФТИ | Москва

Составители: 

Рубрика: 

cÈÏËã


ÒÓˮөËÏÈmÒ°Ò亰ÒmãÒÓˮӺä¹¯º°¯ÈÓ°mË
º°ÈmÒä m©¯ÈÎËÓÒ« ã« °º°mËÓÓºº ÏÓÈËÓÒ« Ò °º°mËÓÓºº mË}º¯ÈmÒ²
º¹¯ËËãËÓÒË
Af f
=
λ
 ºãÈËä
()( )()A u wi i u wi
+=+ +
αβ
ÒãÒ
(
)(
)( )( )Au Aw i u w u w i
+=−++
αβ βα
ÒÒÏ¯ÈmËÓ°mÈË®°mÒËãÓ©²ÒäÓÒ䩲
È°Ë®ÓȲºÒäº
Au u w
Aw u w
=−
=+
αβ
βα

sºªºÒºÏÓÈÈË º º¹Ë¯Èº¯
A
ÒäËË mä˯ӺË ÒÓmȯÒÈÓÓºË ¹º¹¯º
°¯ÈÓ°mº°ºm¹ÈÈËË°ãÒÓˮӺ®ººãº}º®ªãËäËÓºm
u
Ò
w
¹º°}ºã}
()

()()
()()
Au w Au Aw u w u w
uw
ξ
η
ξ
η
ξ
αβ ηβα
ξ
αηβ ηα
ξ
β
+= + = −+ +
=+ +−

˺¯ËäÈº}ÈÏÈÓÈ
~ÈÈÈ

cËËÓÒË
Ëjpzq xvixzknttn otj·ntq¹ q xvixzknttn knrzvé vwnéjzvéj
A

lnpxzkyínmvkwévxzéjtxzknzén}unét}xzvsi|vkqojljttvmvujz
éq|np
−−
−−
−−
122
212
323

°
cÈ°°äº¯Òä °ÓÈÈãÈ °ãÈ® }ºÈº¹Ë¯Èº¯
A
Ë®°mË m }ºä¹ãË}°ÓºäãÒÓˮӺä
¹¯º°¯ÈÓ°mË
rËäÒ°}È°º°mËÓÓ©ËÏÓÈËÓÒ«¹ºÁº¯äãÈä{º°¹ºãϺmÈmÒ°
¹¯ÈmÒãºä¯ÈÏãºÎËÓÒ«º¹¯ËËãÒËã«¹º¹Ë¯mº®°¯º}Ë°ä˺¯Ëä¹ºãÒä
det ( )( ) ( ) ( )
()().
−− −
−−−
−−−
=− + − + − + + − − =
=− + − + =− + −
122
21 2
323
1 1 22 6 6 24 3 3
111
2
32 2
λ
λ
λ
λλ λ λ
λλλ λ λ
|}È¹ºãÈËäºÒÏ¯Ë²°º°mËÓÓ©²ÏÓÈËÓÒ®ºÓº
λ
1
1=
mËË°mËÓÓºËÒ
mÈ
λ
2
= i
Ò
λ
2
=−i
}ºä¹ãË}°Óº°º¹¯«ÎËÓÓ©Ë


vä¹¯ÒãºÎËÓÒË
cÈÏËã 
ÒÓˮөËÏÈmÒ°Ò亰ˆÒmãÒÓˮӺ乯º°ˆ¯ÈÓ°ˆmË



           º°ˆÈmÒä m©¯ÈÎËÓÒ« ã« °º­°ˆmËÓÓºº ÏÓÈËÓÒ« Ò °º­°ˆmËÓÓºº mË}ˆº¯È m Ò²
           º¹¯ËËãËÓÒË                    A f = λ f            ºã‚ÈËä                  A (u + wi ) = (α + β i )(u + wi )                  ÒãÒ
              ) + ( Aw
           ( Au       )i = (α u − β w) + ( β u + α w)i ÒÒϯÈmËÓ°ˆmÈË®°ˆm҈Ëã Ó©²ÒäÓÒ䩲
                                       =αu− β w
                                     Au
           È°ˆË®ÓȲºÒ䈺                      
                                      
                                     Aw = β u + α w
           
           sº ªˆº Ò ºÏÓÈÈˈ ˆº º¹Ë¯Èˆº¯ A  ÒäËˈ m‚ä˯ӺË ÒÓmȯÒÈӈӺË ¹º¹¯º
           °ˆ¯ÈÓ°ˆmº°ºm¹ÈÈ ËË°ãÒÓˮӺ®º­ºãº}º®ªãËäËӈºmuÒw¹º°}ºã }‚
           
                                       A (ξ u + η w) = ξ Au
                                                           + η Aw
                                                                 = ξ (αu − βw) + η ( βu + αw)
                                                           = (ξ α + η β )u + (ηα − ξβ ) w                                     
                                
      ‘˺¯ËäȺ}ÈÏÈÓÈ
               
               
               
 ~ÈÈÈ                  Ëjpzq xvixzkntt€n otj·ntq¹ q xvixzkntt€n knrzvé€ vwnéjzvéj A 
                   lnpxzkyínmv k wévxzéjtxzkn zén}unét€} xzvsi|vk q ojljttvmv ujz
 
                                        −1 − 2 2
                         éq|np         − 2 − 1 2 
 
                                        −3 −2 3
 
 
 cËËÓÒË
               
               
°cÈ°°äºˆ¯Òä °ÓÈÈãÈ °ã‚È® }ºÈ º¹Ë¯Èˆº¯ A  Ë®°ˆm‚ˈ m }ºä¹ãË}°Óºä ãÒÓˮӺä
     ¹¯º°ˆ¯ÈÓ°ˆmË

     r‚ËäÒ°}Ȉ °º­°ˆmËÓÓ©ËÏÓÈËÓÒ«¹ºÁº¯ä‚ãÈä    {º°¹ºã ϺmÈm Ò° 
     ¹¯ÈmÒãºä¯ÈÏãºÎËÓÒ«º¹¯ËËã҈Ë㫹º¹Ë¯mº®°ˆ¯º}Ë °äˆËº¯Ëä‚ ¹ºã‚Òä
                                                
                                                
                   −1− λ              −2    2
               det  −2               −1− λ  2   = − (1 + λ )( λ − 1) 2 + 2(2λ − 6 + 6) + 2(4 − 3 − 3λ ) =
                                                                                                          
                    −3                −2   3− λ
                                                             = − λ3 + λ2 − λ + 1 = −( λ2 + 1)( λ − 1) .
                                                   
                                                   
       |ˆ}‚ȹºã‚ÈË䈺Òψ¯Ë²°º­°ˆmËÓÓ©²ÏÓÈËÓÒ®ºÓº λ1 = 1 m˝˰ˆmËÓÓºËÒ
       mÈ λ2 = i Ò λ2 = −i }ºä¹ãË}°Óº°º¹¯«ÎËÓÓ©Ë 
       


    v乯ÒãºÎËÓÒË

Страницы