Аналитическая геометрия и линейная алгебра. Умнов А.Е. - 281 стр.

UptoLike

ВУЗ: 

МФТИ | Москва

Составители: 

Рубрика: 

cÈÏËã


¯Ò}ãÈÓ©ËÏÈÈÒãÒÓˮӺ®ÈãË¯©
°vºm¹ÈËÓÒË Áº¯äã ÒÏäËÓËÓÒ« äÈ¯Ò }mȯÈÒÓºº ÁÓ}ÒºÓÈãÈ Ò
°È亰º¹¯«ÎËÓÓºº º¹Ë¯Èº¯È ¹¯Ò ¹Ë¯Ë²ºË º ºÓºº º¯ºÓº¯äÒ¯ºmÈÓÓºº
ÈÏÒ°È } ¯ºä ¹ºÏmºã«Ë Ò°¹ºãϺmÈm}ÈË°mË ÈÏÒ°È
},...,,{
21
n
eee
′′′

º¯ºÓº¯äÒ¯ºmÈÓÓ©®ÈÏÒ°ÒÏ°º°mËÓÓ©²mË}º¯ºmº¹Ë¯Èº¯È
ˆ
wºÈ
ÏÒ°°Ë°mË°ä˺¯ËäÒmÓËääÈ¯ÒÈº¹Ë¯Èº¯È
ˆ
È ÏÓÈÒ
Ò äÈ¯ÒÈ }mȯÈÒÓºº ÁÓ}ÒºÓÈãÈ
Φ
()
x
 ÒäËË ÒȺÓÈãÓ©® mÒ
¹¯ÒËäÓÈãÈmÓº®ÒȺÓÈãÒ¯È°¹ºãºÎËÓ©°º°mËÓÓ©ËÏÓÈËÓÒ«°È亰º
¹¯«ÎËÓÓººº¹Ë¯Èº¯È
ˆ

˺¯ËäÈº}ÈÏÈÓÈ
~ÈäËÒäºm˯ÎËÓÒË˺¯Ëä©°ºãÈ°Ë°«°m˯ÎËÓÒËä°ãË°mÒ«

j°¹ºãÏ« ˺¯Ëä  äºÎÓº ¹¯º°Ò¹¯ºË¯ ºËÓ}Ò ª}°¯ËäÈãÓ©²
ÏÓÈËÓÒ®}mȯÈÒÓººÁÓ}ÒºÓÈãÈ{}ÈË°mË¹¯Òä˯È¯È°°äº¯ÒäÏÈÈÓȲºÎ
ËÓÒ«äÈ}°ÒääÈÒäÒÓÒääÈã«ºÓºËÓÒ«cËãË«
|¹¯ËËãËÓÒË

nÓ}ÒºÓÈã
ρ
()
(,
)
(,)
x
xAx
xx
=
ÏÈÈÓÓ©®m
E
n
ã«ÓË}ºº¯ºº°È亰º¹¯«
ÎËÓÓººº¹Ë¯Èº¯È
A
ÓÈÏ©mÈË°«vztv¡ntqnuènsn¹
vãË°mÒË

{º¯ºÓº¯äÒ¯ºmÈÓÓºäÈÏÒ°ËäÈ}°ÒäÈãÓºËäÒÓÒäÈãÓºËÏÓÈËÓÒË
ºÓºËÓÒ«cËãË«¯ÈmÓºäÈ}°ÒäÈãÓºääÒÓÒäÈãÓºä°º°mËÓÓº
äÏÓÈËÓÒº¹Ë¯Èº¯È
A
ÒªºÏÓÈËÓÒËº°ÒÈË°«ÓÈ°ººmË°
mËä°º°mËÓÓºämË}º¯Ëªººº¹Ë¯Èº¯È
iº}ÈÏÈËã°mº
º°}ºã}¹¯Ò¹ ˯˲ºË}º¯ºÓº¯äÒ¯ºmÈÓÓºäÈÏÒ° º¯ÈϺmÈÓÓºäÒÏ°º
°mËÓÓ©² mË}º¯ºm °È亰º¹¯«ÎËÓÓºº º¹Ë¯Èº¯È
A
m °Òã ˺¯Ëä© 
°¹¯ÈmËãÒm©°ººÓºËÓÒ«
∑
∑
∑
∑
=
=
=
=
′
′
===
n
i
i
n
i
ii
n
i
i
n
i
jiij
xx
xAx
x
1
1
1
1
),(
)
ˆ
,(
)(
2
2
2
ξ
ξ
λ
ξ
ξ
ξ
α
ρ

º¹¯ºmº«¯È°°ÎËÓÒ«ÈÓÈãºÒÓ©Ë º}ÈÏÈËã°m˺¯Ëä© ¹ºãÈËä
º
λ
ρ
λ
min max
()
≤≤
x

vãË°mÒËº}ÈÏÈÓº
cÈÏËã
¯Ò}ãÈÓ©ËÏÈÈÒãÒÓˮӺ®ÈãË­¯©



           °vºm¹ÈËÓÒË Áº¯ä‚ã ÒÏäËÓËÓÒ« äȈ¯Ò }mȯȈÒÓºº Á‚Ó}ÒºÓÈãÈ Ò
                °È亰º¹¯«ÎËÓÓºº º¹Ë¯Èˆº¯È ¹¯Ò ¹Ë¯Ë²ºË ºˆ ºÓºº º¯ˆºÓº¯äÒ¯ºmÈÓÓºº
                ­ÈÏÒ°È } ¯‚ºä‚ ¹ºÏmºã«Ëˆ Ò°¹ºã ϺmȈ  m }ÈË°ˆmË ­ÈÏÒ°È {e1′ , e 2′ ,..., e n′ }  
                                                                                                                                ˆ
                   º¯ˆºÓº¯äÒ¯ºmÈÓÓ©®­ÈÏÒ°ÒÏ°º­°ˆmËÓÓ©²mË}ˆº¯ºmº¹Ë¯Èˆº¯È  wˆºˆ­È
                                                                                                                                 ˆ
                   ÏÒ°°‚Ë°ˆm‚ˈ °äˆËº¯Ëä‚ ÒmÓËääȈ¯ÒȺ¹Ë¯Èˆº¯È   ÈÏÓÈ҈
                   Ò äȈ¯ÒÈ }mȯȈÒÓºº Á‚Ó}ÒºÓÈãÈ Φ ( x )  ÒäËˈ ÒȺÓÈã Ó©® mÒ
                   ¹¯ÒËäÓÈãÈmÓº®ÒȺÓÈãүȰ¹ºãºÎËÓ©°º­°ˆmËÓÓ©ËÏÓÈËÓÒ«°È亰º
                                                           ˆ
           ¹¯«ÎËÓÓººº¹Ë¯Èˆº¯È  
           
           
    ‘˺¯ËäȺ}ÈÏÈÓÈ
         
         
        ~ÈäˈÒ䈺‚ˆm˯ÎËÓÒˈ˺¯Ëä©°ºãÈ°‚ˈ°«°‚ˆm˯ÎËÓÒËä°ãË°ˆmÒ«

         
         
         
         j°¹ºã ς« ˆËº¯Ëä‚  äºÎÓº ‚¹¯º°ˆÒˆ  ¹¯ºË‚¯‚ ºËÓ}Ò ª}°ˆ¯ËäÈã Ó©²
ÏÓÈËÓÒ®}mȯȈÒÓººÁ‚Ó}ÒºÓÈãÈ{}ÈË°ˆm˹¯Òä˯ȯȰ°äºˆ¯ÒäÏÈÈ‚ÓȲºÎ
ËÓÒ«äÈ}°Òä‚äÈÒäÒÓÒä‚äÈ㫺ˆÓº ËÓÒ«cËãË«
         
         
                                                   )
                                             ( x , Ax
    |¹¯ËËãËÓÒË          n‚Ó}ÒºÓÈã ρ ( x ) =        ÏÈÈÓÓ©®m E n ã«ÓË}ºˆº¯ºº°È亰º¹¯«
                                      ( x, x)
                          ÎËÓÓººº¹Ë¯Èˆº¯È A ÓÈÏ©mÈˈ°«vztv¡ntqnuènsn¹
               
               
    vãË°ˆmÒË            {º¯ˆºÓº¯äÒ¯ºmÈÓÓºä­ÈÏÒ°ËäÈ}°ÒäÈã ÓºË äÒÓÒäÈã ÓºË ÏÓÈËÓÒË
                  ºˆÓºËÓÒ«cËãË«¯ÈmÓºäÈ}°ÒäÈã Óºä‚ äÒÓÒäÈã Óºä‚ °º­°ˆmËÓÓº
                          ä‚ÏÓÈËÓÒ º¹Ë¯Èˆº¯È A ÒªˆºÏÓÈËÓÒ˺°ˆÒÈˈ°«ÓÈ°ººˆmˈ°ˆ
                          m‚ Ëä°º­°ˆmËÓÓºämË}ˆº¯Ëªˆººº¹Ë¯Èˆº¯È
           
     iº}ÈÏȈËã°ˆmº
      
         º°}ºã }‚¹¯Ò¹Ë¯Ë²ºË}º¯ˆºÓº¯äÒ¯ºmÈÓӺ䂭ÈÏÒ°‚º­¯ÈϺmÈÓÓºä‚ÒÏ°º­
           °ˆmËÓÓ©² mË}ˆº¯ºm °È亰º¹¯«ÎËÓÓºº º¹Ë¯Èˆº¯È A  m °Òã‚ ˆËº¯Ëä©  
           °¹¯ÈmËãÒm©°ººˆÓº ËÓÒ«
                                                                              n                       n

                                                           ( x, Aˆ x )
                                                                            ∑ α ij ξ iξ j ∑ λi ξ i′2
                                                                            i =1                     i =1
                                                  ρ ( x) =             =              n
                                                                                                 =      n
                                                                                                                   
                                                            ( x, x )
                                                                                  ∑       ξ i2        ∑ ξ i′   2

                                                                                  i =1                i =1
                                                                                  

         ˆº ¹¯ºmº« ¯È°°‚ÎËÓÒ« ÈÓÈãºÒÓ©Ë º}ÈÏȈËã °ˆm‚ ˆËº¯Ëä©  ¹ºã‚ÈËä
         ˆº λmin ≤ ρ ( x ) ≤ λmax 
              
      vãË°ˆmÒ˺}ÈÏÈÓº
           

Страницы