Аналитическая геометрия и линейная алгебра. Умнов А.Е. - 208 стр.

UptoLike

ВУЗ: 

МФТИ | Москва

Составители: 

Рубрика: 


Ë} ÒÒ}ÈÁË ¯©m©° Ë®äÈËäÈÒ}Òlnj
ÙkÓÈãÒÒË°}È«˺äË¯Ò«ÒãÒÓË®ÓÈ«ÈãË¯Èµ

äÓºmkp
iº}ÈÏÈËã°mº
~ÈäËÒäºº¹Ë¯Èº¯
EA
ˆ
ˆ
λ
−
ºËmÒÓºãÒÓˮө®m°ÒããÒÓˮӺ°Òº¹Ë¯Èº
¯ºm
A
ˆ
Ò
E
ˆ
ºÈ°ºãÈ°Óº°ãË°mÒ˺º¹¯ËËãÒËãÓËäËÓ«Ë°«¹¯Ò
ÏÈäËÓË ÈÏÒ°È ºªºä ¹¯Ò ¹Ë¯Ë²ºË º ÈÏÒ°È
{ , ,..., }gg g
n
12
}ÈÏÒ°
{, ,..., }
′′ ′
gg g
n
12
ÒäËËä
.
ˆ
ˆ
det
ˆ
ˆ
det
gg
EAEA
λλ
−=−
′
˺¯ËäÈº}ÈÏÈÓÈ
˺¯ËäÈ

vºmº}¹Óº°°º°mËÓÓ©²mË}º¯ºm ºmËÈÒ²ÓË}ºº¯ºä°º°
mËÓÓºä ÏÓÈËÓÒ ãÒÓˮӺº º¹Ë¯Èº¯È
A
 º¹ºãÓËÓÓÈ« ÓãËm©ä
ªãËäËÓºä ãÒÓˮӺ
º ¹¯º°¯ÈÓ°mÈ
Λ
 «mã«Ë°« ÒÓmȯÒÈÓÓ©ä ¹º
¹¯º°¯ÈÓ°mºäº¹Ë¯Èº¯È
A
iº}ÈÏÈËã°mº
°
11
ˆ
ffA
λ
=
Ò
22
ˆ
ffA
λ
=
ºÈã«ã©²ÓË¯ÈmÓ©²ÓãºÓºm¯ËäËÓÓºÒ
°Ëã
α
Ò
β
)(
ˆˆ
)(
ˆ
21212121
fffffAfAffA
βαλλβλαβαβα
+=+=+=+

ºÒ¹º}ÈÏ©mÈË°¹¯ÈmËãÒmº°m˯ÎËÓÒ«˺¯Ëä©
˺¯ËäÈº}ÈÏÈÓÈ
|¹¯ËËãËÓÒË

º¹¯º°¯ÈÓ°mº °º°º«ËË ÒÏ °º°mËÓÓ©² mË}º¯ºm ºmËÈÒ²
ÓË}ºº¯ºä°º°mËÓÓºäÏÓÈËÓÒº¹ºãÓËÓÓ©²ÓãËm©äªãËäËÓºä
ÓÈÏ©mÈË°«qtkjéqjtztuxvixzknttuwvlwévxzéjtxzkvuãÒÓË®
Óººº¹Ë¯Èº¯È
A

vãË°mÒË

{°«}ºËÒÓmȯÒÈÓÓºË°º°mËÓÓºË¹º¹¯º°¯ÈÓ°mºãÒÓˮӺºº¹Ë¯È
º¯È
$
«mã«Ë°«È}ÎË ÒÓmȯÒÈÓÓ©ä ¹º¹¯º°¯ÈÓ°mºä ãÒÓˮӺº
º¹Ë¯Èº¯È
%
Ë°ãÒº¹Ë¯Èº¯©
$
Ò
%
}ºääÒ¯
iº}ÈÏÈËã°mº
°
∗
Λ
ÒÓmȯÒÈÓÓºË°º°mËÓÓºË¹º¹¯º°¯ÈÓ°mº º¹Ë¯Èº¯È
A
ºË°
∗
Λ∈∀=
fffA ,
ˆ
λ

 Ë }  Ò Ò    } È Á Ë  ¯ ©   m © °  Ë ®   ä È ˆ Ë ä È ˆ Ò } Ò   l n ‘ j 
ÙkÓÈã҈ÒË°}È«˺äˈ¯Ò«ÒãÒÓË®ÓÈ«ÈãË­¯ÈµäÓºmkp



  iº}ÈÏȈËã°ˆmº
         
          ~ÈäˈÒ䈺º¹Ë¯Èˆº¯ Aˆ − λ Eˆ ºËmÒÓºãÒÓˮө®m°Òã‚ãÒÓˮӺ°ˆÒº¹Ë¯Èˆº
          ¯ºm Â Ò Ê ‘ºÈ°ºãÈ°Óº°ãË°ˆmÒ Ëºº¹¯ËËã҈Ëã ÓËäËӫˈ°«¹¯Ò
          ÏÈäËÓË ­ÈÏÒ°È ºªˆºä‚ ¹¯Ò ¹Ë¯Ë²ºË ºˆ ­ÈÏÒ°È {g1 , g 2 ,..., g n }  } ­ÈÏÒ°‚
           {g1′ , g 2′ ,..., g n′ } ÒäËËä
                                                     det Aˆ − λ Eˆ            = det Aˆ − λ Eˆ         .
                                                                         g′                       g
     ‘˺¯ËäȺ}ÈÏÈÓÈ
              
              
              
 ‘˺¯ËäÈ               vºmº}‚¹Óº°ˆ  °º­°ˆmËÓÓ©² mË}ˆº¯ºm ºˆmËÈ Ò² ÓË}ºˆº¯ºä‚ °º­°ˆ
 
                        mËÓÓºä‚ ÏÓÈËÓÒ  ãÒÓˮӺº º¹Ë¯Èˆº¯È A  º¹ºãÓËÓÓÈ« ӂãËm©ä
                        ªãËäËӈºä ãÒÓˮӺº ¹¯º°ˆ¯ÈÓ°ˆmÈ Λ  «mã«Ëˆ°« ÒÓmȯÒÈӈөä ¹º
                        ¹¯º°ˆ¯ÈÓ°ˆmºäº¹Ë¯Èˆº¯È A 
        
  iº}ÈÏȈËã°ˆmº
   
         ‚°ˆ Aˆ f1 = λ f1 Ò Aˆ f 2 = λ f 2 ‘ºÈã«ã ­©²Ó˯ÈmÓ©²ӂã ºÓºm¯ËäËÓÓºÒ
         °Ëã α Ò β 
                                Aˆ (α f1 + β f 2 )= α Aˆ f1 + β Aˆ f 2 = α λ f1 + β λ f 2 = λ (α f1 + β f 2 ) 
         
         ˆºÒ¹º}ÈÏ©mÈˈ°¹¯ÈmËãÒmº°ˆ ‚ˆm˯ÎËÓÒ«ˆËº¯Ëä©
     
     ‘˺¯ËäȺ}ÈÏÈÓÈ
          
          
          
 |¹¯ËËãËÓÒË            º¹¯º°ˆ¯ÈÓ°ˆmº °º°ˆº«ËË ÒÏ °º­°ˆmËÓÓ©² mË}ˆº¯ºm ºˆmËÈ Ò²
                  ÓË}ºˆº¯ºä‚°º­°ˆmËÓÓºä‚ÏÓÈËÓÒ º¹ºãÓËÓÓ©²ӂãËm©äªãËäËӈºä
                         ÓÈÏ©mÈˈ°« qtkjéqjtzt€uxvixzkntt€uwvlwévxzéjtxzkvuãÒÓË®
                         Óººº¹Ë¯Èˆº¯È A 
          
          
          
 vãË°ˆmÒË             {°«}ºËÒÓmȯÒÈӈӺË°º­°ˆmËÓӺ˹º¹¯º°ˆ¯ÈÓ°ˆmºãÒÓˮӺºº¹Ë¯È
                         «mã«Ëˆ°« ˆÈ}ÎË ÒÓmȯÒÈӈөä ¹º¹¯º°ˆ¯ÈÓ°ˆmºä ãÒÓˮӺº
                        ˆº¯È $
                                    Ë°ãÒº¹Ë¯Èˆº¯© $ Ò % }ºä䂈ү‚ ˆ
                        º¹Ë¯Èˆº¯È %
        
  iº}ÈÏȈËã°ˆmº
      
      
          ‚°ˆ  Λ∗   ÒÓmȯÒÈӈӺË °º­°ˆmËÓÓºË ¹º¹¯º°ˆ¯ÈÓ°ˆmº º¹Ë¯Èˆº¯È A  ˆº Ë°ˆ 
           Aˆ f = λf , ∀f ∈Λ∗ 
          

Страницы