Аналитическая геометрия и линейная алгебра. Умнов А.Е. - 228 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
°° }mȯÈÒÓ©® ÁÓ}ÒºÓÈã
)(x
ÒäËË m ÓË}ºº¯ºä ÈÏÒ°Ë
},...,,{
21
n
ggg
¹¯Ë°ÈmãËÓÒË
)(x
∑∑
==
=
n
i
n
j
jiij
11
ξ
ξ
ϕ
Ò¹° °Ë°m mÈ
¯ÈÏãÒÓ©²ÈÏÒ°È
{, ,..., }
′′ ′
gg g
n
12
Ò
},...,,{
21
n
ggg
′′′′′′
m}ºº¯©²
Φ
(x)
ÒäËË°ãË
Ò®mÒ
)(x
],1[,0;;
1
2
1
2
minm
i
m
ki
ii
k
i
ii
=>∀≤−=
∑∑
+==
ληληλ
Ò°ººmË°mËÓÓº
)(x
.],1[,0;;
1
2
1
2
qinq
i
q
pi
ii
p
i
ii
=>∀≤−=
∑∑
+==
µκµκµ
{ °Òã °ËãÈÓÓ©² ¹¯Ë¹ºãºÎËÓÒ® ºãÎÓ© °Ë°mºmÈ ÓËm©¯ºÎËÓÓ©Ë
äÈ¯Ò© ÏÈäËÓ© ¹Ë¯ËäËÓÓ©²
ω
ij
Ò
θ
ij
¹¯Ò ¹Ë¯Ë²ºÈ² º ÈÏÒ°È
},...,,{
21
n
ggg
}ÈÏÒ°Èä
},...,,{
21
n
ggg
′′′
Ò
},...,,{
21
n
ggg
′′′′′′
È}ÒËº

],1[;
1
ns
n
j
jsjs
==
∑
=
ξ
ωη
Ò
],1[;
1
ns
n
j
jsjs
==
∑
=
ξ
θκ

°¯Ò¯ÈmÓ«Ëä ÏÓÈËÓÒ« ÁÓ}ÒºÓÈãÈ
)(
x
m ÈÏҰȲ
},...,,{
21
n
ggg
′′′
Ò
},...,,{
21
n
ggg
′′′′′′
ã«ÓË}ºº¯ººªãËäËÓÈ
j
n
j
ji
n
i
ik
n
k
k
gggx
′′
=
′
==
∑∑∑
===
111
κη
ξ
∑∑∑∑
+==+==
−=−
q
pi
ii
p
i
ii
m
ki
ii
k
i
ii
1
2
1
2
1
2
1
2
κµκµηληλ
Ò¹¯Ëº¯ÈÏËä¹ºãËÓÓºË¯ÈmËÓ°mº}mÒ

∑∑∑∑
+==+==
+=+
m
ki
ii
p
i
ii
q
pi
ii
k
i
ii
1
2
1
2
1
2
1
2
ηλκµκµηλ

°j°°ãËËä¹ºãËÓÓºË°ººÓºËÓÒË
iº¹°Òäº
pk
<
Ò¹¯Ë¹ºãºÎÒäºªãËäËÓ
x
ÒäËËm¯È°°äÈ¯ÒmÈË䩲
ÈÏҰȲ }ºä¹ºÓËÓ©
],1[,0;],1[,0 npiki
ii
+=∀==∀=
κη
 wÒ² °ãºmÒ®
äËÓË Ëä
n
 ¹º°}ºã}
pk
<
 p°ãÒ Ò² ¹º°ÈmÒm¯ÈmËÓ°mÈº
ä©¹ºãÒäºÓº¯ºÓ°Ò°ËäãÒÓˮө²¯ÈmÓËÓÒ®ºÓº°ÒËãÓºÓËÒÏ
mË°Ó©²
},...,,{
21
n
ξ
ξ
ξ
º°}ºã}Ò°ãº È}Ò²¯ÈmÓËÓÒ®äËÓËÒ°ãÈ ÓË
 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
    
         °‚°ˆ  }mȯȈÒÓ©® Á‚Ó}ÒºÓÈã  (x )  ÒäËˈ m ÓË}ºˆº¯ºä ­ÈÏÒ°Ë
                                                                                        n     n
                 {g1, g 2 ,..., g n }  ¹¯Ë°ˆÈmãËÓÒË  (x ) = ∑ ∑ ϕ ijξ iξ j  Ò ¹‚°ˆ  °‚Ë°ˆm‚ ˆ mÈ
                                                                                       i =1 j =1
                 ¯ÈÏãÒÓ©²­ÈÏÒ°È {g1′ , g 2′ ,..., g n′ } Ò {g1′′, g 2′′ ,..., g n′′ } m}ºˆº¯©² Φ(x)ÒäËˈ°ãË
                 ‚ Ò®mÒ
                                                   k                    m
                                       (x ) = ∑ λ i ηi2 −             ∑ λ iηi2 ;            m ≤ n ; ∀λ i > 0 , i = [1, m] 
                                                  i =1               i = k +1
                 
                 Ò°ººˆmˈ°ˆmËÓÓº
                 
                                                   p                    q
                                       (x ) = ∑ µ i κ i2 −            ∑ µi κ i2 ;            q ≤ n ; ∀µ i > 0 , i = [1, q] . 
                                                 i =1                i = p +1
                                                      
                 { °Òã‚ °ËãÈÓÓ©² ¹¯Ë¹ºãºÎËÓÒ® ºãÎÓ© °‚Ë°ˆmºmȈ  ÓËm©¯ºÎËÓÓ©Ë
                 äȈ¯Ò© ÏÈäËÓ© ¹Ë¯ËäËÓÓ©²                                   ωij  Ò θij  ¹¯Ò ¹Ë¯Ë²ºȲ ºˆ ­ÈÏÒ°È
                 {g1, g 2 ,..., g n } }­ÈÏÒ°Èä{g1′ , g 2′ ,..., g n′ } Ò{g1′′, g 2′′ ,..., g n′′ } ˆÈ}Òˈº
                 
                                           n                                                             n
                  η s =    ∑ω sjξ j ; s = [1, n] Ò κ s = ∑θ sjξ j ; s = [1, n]   
                                          j =1                                                           j =1
                
           °¯Ò¯ÈmÓ«Ëä ÏÓÈËÓÒ« Á‚Ó}ÒºÓÈãÈ  (x )                                                       m ­ÈÏҰȲ          {g1′ , g 2′ ,..., g n′ }  Ò
                                                                                                     n                      n        n
                 {g1′′, g 2′′ ,..., g n′′ } ã«ÓË}ºˆº¯ººªãËäËӈÈ x = ∑ ξ k g k = ∑ηi g i′ = ∑ κ j g ′′j 
                                                                                                   k =1                    i =1     j =1
                 

                                                   k                    m                    p                         q
                                                  ∑      λ i ηi2   −   ∑    λ i ηi2    =∑          µ i κ i2     −    ∑ µi κ i2 
                                                  i =1               i = k +1               i =1                    i = p +1
                                                     
                 Ò¹¯Ëº­¯ÈςË乺ã‚ËÓӺ˯ÈmËÓ°ˆmº}mÒ‚
                 
                                                   k                     q                   p                         m
                      ∑      λ i ηi2   +   ∑    µ i κ i2   =∑          µ i κ i2     +     ∑ λ iηi2   
                                                  i =1               i = p +1               i =1                    i = k +1
               
           °j°°ãË‚Ë乺ã‚ËÓӺ˰ººˆÓº ËÓÒË
           
               iº¹‚°ˆÒ䈺 k < p Ò¹¯Ë¹ºãºÎÒ䈺ªãËäËӈxÒäËˈm¯È°°äȈ¯ÒmÈË䩲
                 ­ÈÏҰȲ }ºä¹ºÓËӈ© ηi = 0 , ∀i = [1, k ] ; κ i = 0 , ∀i = [ p + 1, n]  wˆÒ² ‚°ãºmÒ®
                 äËÓ Ë Ëä n ¹º°}ºã }‚ k < p  p°ãÒ Ò² ¹º°ˆÈm҈  m ¯ÈmËÓ°ˆmÈ   ˆº
                 䩹ºã‚ÒäºÓº¯ºӂ °Ò°ˆËä‚ãÒÓˮө²‚¯ÈmÓËÓÒ®ºˆÓº°ÒˆËã ÓºÓËÒÏ
                 mË°ˆÓ©² {ξ1 , ξ 2 ,..., ξ n }  º°}ºã }‚ Ұ㺠ˆÈ}Ò² ‚¯ÈmÓËÓÒ® äËÓ Ë Ò°ãÈ ÓË

Страницы