Аналитическая геометрия и линейная алгебра. Умнов А.Е. - 204 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
vmº®°mÈ°º°mËÓÓ©²mË}º¯ºmÒ°º°mËÓÓ©²ÏÓÈËÓÒ®
˺¯ËäÈ

{ }ºä¹ãË}°Óºä ãÒÓˮӺä ¹¯º°¯ÈÓ°mË
n
Λ
m°«}Ò® ãÒÓˮө® º¹Ë¯È
º¯ÒäËË²º«©ºÒÓ°º°mËÓÓ©®mË}º¯
iº}ÈÏÈËã°mº
º°}ºã} ²È¯È}˯ҰÒË°}ºË ¯ÈmÓËÓÒË «mã«Ë°« ÈãË¯ÈÒË°}Òä ¯ÈmÓËÓÒËä
n
® °˹ËÓÒ ºÓº°ÒËãÓº
λ
 º } ÓËä ¹¯ÒäËÓÒäÈ vxtvktj¹ znvénuj kx¡np
jsmnié
m˯ÎÈÈ«ºÈ}ºË¯ÈmÓËÓÒËÒäËË²º«©ºÒÓ}ºä¹ãË}°
Ó©®}º¯ËÓ
˺¯ËäÈº}ÈÏÈÓÈ
{°ãÈËmËË°mËÓÓººãÒÓˮӺº¹¯º°¯ÈÓ°mÈ˺¯ËäÈÓËm˯ÓÈsȹ¯Ò
ä˯ãÒÓˮө®º¹Ë¯Èº¯¹ºmº¯ºÈ¹ãº°}º°Ò
Oxy
mº}¯ÓÈÈãÈ}ºº¯ÒÓÈÓÈºã
ϕ
≠k
π
ÓËÒäËËÓÒºÓºº°º°mËÓÓººmË}º¯ÈiË®°mÒËãÓº²È¯È}˯ҰÒË°}ºË¯ÈmÓË
ÓÒËã«ªººº¹Ë¯Èº¯ÈÒäËËmÒ°ä¹
det
cos sin
sin cos
ϕλ ϕ
ϕϕλ
−−
−
= 0
ÒãÒ
λλϕ
2
210−+=cos

ºË°
λϕ ϕ
=±
cos sin
i
|°È°ãËËº¹¯Ò
ϕ
≠k
π
mËË°mËÓÓ©²¯ËËÓÒ®ÈÓÓºË
²È¯È}˯ҰÒË°}ºË¯ÈmÓËÓÒËÓËÒäËË
˺¯ËäÈ

{mËË°mËÓÓºäãÒÓˮӺä¹¯º°¯ÈÓ°mË
n
Λ
m°«}Ò®ãÒÓˮө®º¹Ë¯È
º¯ ÒäËË ãÒº ²º« © ºÒÓ °º°mËÓÓ©® mË}º¯ ãÒº mä˯ӺË
ÒÓmȯÒÈÓÓºË¹º¹¯º°¯ÈÓ°mº
iº}ÈÏÈËã°mº
p°ãÒ²È¯È}˯ҰÒË°}ºË¯ÈmÓËÓÒËÒäËËmËË°mËÓÓ©®}º¯ËÓºÒÏ°Ò°Ëä©
ÓȲºÒä°º°mËÓÓ©®mË}º¯
° ²È¯È}˯ҰÒË°}ºË ¯ÈmÓËÓÒË ÒäËË }ºä¹ãË}°Ó©® }º¯ËÓ
λαβ
=+i

¯ËÒm°Ò°Ëä¹ºãÒä°ººmË°mÒ®Ëä}ºä¹ãË}°ÓºÏÓÈÓ©®°º
°mËÓÓ©® mË}º¯
fuwi=+
 Ë
u
Ò
w
 ªãËäËÓ©
n
Λ
 ¹¯Ë°Èmã«Ëä©Ë
mËË°mËÓÓ©äÒ
n
}ºä¹ºÓËÓÓ©äÒ°ºãÈäÒ
º}ÈÎËäº
u
Ò
w
ãÒÓˮӺÓËÏÈmÒ°Òä©Ëiº¹°Òä¹¯ºÒmÓºË
uw=
κ
ºÈÒÏ
°ººÓºËÓÒ«
Af f
=
λ
ÒäËËä º
(( ) ) ( )Aiw iw
κλκ
+=+
 ÒãÒ
Aw w
=
λ
 º
Ë°
λ
mËË°mËÓÓºË º ¹¯ºÒmº¯ËÒ ¹¯Ë¹ºãºÎËÓÒ º ÓËmËË°mËÓÓº°Ò
°º°mËÓÓººÏÓÈËÓÒ«

iº}ÈÏ©mÈË°«Óȹ¯Òä˯m}¯°Ënz
 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



vmº®°ˆmÈ°º­°ˆmËÓÓ©²mË}ˆº¯ºmÒ°º­°ˆmËÓÓ©²ÏÓÈËÓÒ®



 ‘˺¯ËäÈ               { }ºä¹ãË}°Óºä ãÒÓˮӺä ¹¯º°ˆ¯ÈÓ°ˆmË Λn  m°«}Ò® ãÒÓˮө® º¹Ë¯È
                 ˆº¯ÒäËˈ²ºˆ«­©ºÒÓ°º­°ˆmËÓÓ©®mË}ˆº¯
              
  iº}ÈÏȈËã°ˆmº
      
      º°}ºã }‚ ²È¯È}ˆË¯Ò°ˆÒË°}ºË ‚¯ÈmÓËÓÒË «mã«Ëˆ°« ÈãË­¯ÈÒË°}Òä ‚¯ÈmÓËÓÒËä
      n® °ˆË¹ËÓÒ ºˆÓº°ÒˆËã Óº λ ˆº } ÓËä‚ ¹¯ÒäËÓÒäÈ vxtvktj¹ znvénuj k€x¡np
      jsmnié€  ‚ˆm˯ÎÈ È«ˆºˆÈ}ºË‚¯ÈmÓËÓÒËÒäËˈ²ºˆ«­©ºÒÓ}ºä¹ãË}°
      Ó©®}º¯ËÓ 
      
   ‘˺¯ËäȺ}ÈÏÈÓÈ
      
      
      {°ã‚ÈËm˝˰ˆmËÓÓººãÒÓˮӺº¹¯º°ˆ¯ÈÓ°ˆmȈ˺¯ËäÈÓËm˯ÓÈsȹ¯Ò
ä˯ãÒÓˮө®º¹Ë¯Èˆº¯¹ºmº¯ºˆÈ¹ãº°}º°ˆÒ Oxy mº}¯‚ÓÈÈãÈ}ºº¯ÒÓȈÓÈ‚ºãϕ≠kπ
ÓËÒäËˈÓÒºÓºº°º­°ˆmËÓÓººmË}ˆº¯ÈiË®°ˆm҈Ëã Óº²È¯È}ˆË¯Ò°ˆÒË°}ºË‚¯ÈmÓË
ÓÒË㫪ˆººº¹Ë¯Èˆº¯ÈÒäËˈmÒ °ä¹ 
          

                                      cos ϕ − λ         − sin ϕ
                               det                               = 0 ÒãÒ λ 2 − 2λ cos ϕ + 1 = 0 
                                        sin ϕ          cos ϕ − λ
          

ˆºË°ˆ  λ = cos ϕ ± i sin ϕ |ˆ° È°ãË‚ˈˆº¹¯Ò ϕ≠kπm˝˰ˆmËÓÓ©²¯Ë ËÓÒ®ÈÓÓºË
²È¯È}ˆË¯Ò°ˆÒË°}ºË‚¯ÈmÓËÓÒËÓËÒäËˈ
      
      
 ‘˺¯ËäÈ               {m˝˰ˆmËÓÓºäãÒÓˮӺ乯º°ˆ¯ÈÓ°ˆmË Λn m°«}Ò®ãÒÓˮө®º¹Ë¯È
                 ˆº¯ ÒäËˈ ãÒ­º ²ºˆ« ­© ºÒÓ °º­°ˆmËÓÓ©® mË}ˆº¯ ãÒ­º m‚ä˯ӺË
                        ÒÓmȯÒÈӈӺ˹º¹¯º°ˆ¯ÈÓ°ˆmº
              
  iº}ÈÏȈËã°ˆmº
   
      p°ãҲȯÈ}ˆË¯Ò°ˆÒË°}ºË‚¯ÈmÓËÓÒËÒäËˈm˝˰ˆmËÓÓ©®}º¯ËÓ ˆºÒÏ°Ò°ˆËä©
        ÓȲºÒä°º­°ˆmËÓÓ©®mË}ˆº¯
          
          ‚°ˆ  ²È¯È}ˆË¯Ò°ˆÒË°}ºË ‚¯ÈmÓËÓÒË ÒäËˈ }ºä¹ãË}°Ó©® }º¯ËÓ λ = α + β i 
          ¯Ë Òm°Ò°ˆËä‚  ¹ºã‚Òä°ººˆmˈ°ˆm‚ Ò®Ëä‚}ºä¹ãË}°ÓºÏÓÈÓ©®°º­
          °ˆmËÓÓ©® mË}ˆº¯                 f = u + wi  Ë u Ò w  ªãËäËӈ© Λn  ¹¯Ë°ˆÈmã«Ëä©Ë
          m˝˰ˆmËÓÓ©äÒn}ºä¹ºÓËӈөäÒ°ˆºã­ÈäÒ
          

          º}ÈÎË䈺uÒwãÒÓˮӺÓËÏÈmÒ°Òä©Ëiº¹‚°ˆÒ乯ºˆÒmÓºË u = κw ‘ºÈÒÏ
          °ººˆÓº ËÓÒ« A f = λ f  ÒäËËä ˆº A ((κ + i ) w) = λ (κ + i ) w  ÒãÒ A w = λ w  ˆº
         Ë°ˆ  λm˝˰ˆmËÓÓºË ˆº ¹¯ºˆÒmº¯Ë҈ ¹¯Ë¹ºãºÎËÓÒ  º ÓËm˝˰ˆmËÓÓº°ˆÒ
         °º­°ˆmËÓÓººÏÓÈËÓÒ«
         

   iº}ÈÏ©mÈˈ°«Óȹ¯Òä˯m}‚¯°Ë‘nz

Страницы