Аналитическая геометрия и линейная алгебра. Умнов А.Е. - 265 стр.

UptoLike

Составители: 

Рубрика: 

cÈÏËã
 

pm}ãÒºmº¹¯º°¯ÈÓ°mº
(
,
)(

,) (,) ;, [,]
Ae Ae A Ae e e e i j n
ij ij iij iij
====
+
λλδ
1
sºªººÏÓÈÈË º
==
eAein
i
i
i
1
1
λ
;[,]
È}ÎËÈÏÒ°Ò¹¯Òºäº¯ºÓº¯äÒ¯ºmÈÓÓ©®
°
¯ÒäËä ÏÈ Ò°}ºä©® º¯ººÓÈãÓ©® º¹Ë¯Èº¯
Q
 º¹Ë¯Èº¯ ¹Ë¯Ëmº«Ò®
º¯ºÓº¯äÒ¯ºmÈÓÓ©® ÈÏÒ°
{ , ,..., }ee e
n
12
m º¯ºÓº¯äÒ¯ºmÈÓÓ©® ÈÏÒ°
{ , ,..., }
′′
ee e
n
12
ÒËÒä°«ºm}È˰mË
R
äºÎÓºmÏ«º¹Ë¯Èº¯

QA
1

iË®°mÒËãÓº mº¹Ë¯m©² ÒäËË ä˰º ¯ÈmËÓ°mº

AQR
=
 {ºmº¯©² ÒÏ
°ººÓºËÓÒ®
 
;[,]Re Q Ae Q e e i n
i i ii ii
==
==
−−
11
1
λλ
°ãËË º ÈÏÒ°Ó©Ë
ªãËäËÓ©
ei n
i
,[,]= 1
˰ °º°mËÓÓ©Ë mË}º¯© º¹Ë¯Èº¯È
R
 ºmËÈÒË
¹ºãºÎÒËãÓ©ä °º°mËÓÓ©ä ÏÓÈËÓÒ«ä
λ
i
ÈÏÓÈÒäÈ¯ÒÈ
R
e
mÈ
ÏÒ°Ë
{ , ,..., }ee e
n
12
ÒȺÓÈãÓÈ«Ò¹ººä°ÒääË¯Ò˰}È«ºÈm°ÒããËää©
º¹Ë¯Èº¯
R
°È亰º¹¯«ÎËÓÓ©®
°
º}ÈÎËäÓÈ}ºÓËËÒÓ°mËÓÓº° ¯ÈÏãºÎËÓÒ«{º mmËËÓÓ©²ººÏÓÈËÓÒ«²
°¹¯ÈmËãÒmº¯ÈmËÓ°mº

AA R
+
=
2
¹º°}ºã}ÒÏ

AQR=
Ò
ARQ
+++
=
°ãËË
º





AA RQQR RQ QR RR
+++ + +
== =
1

ºm°Òã°È亰º¹¯«ÎËÓÓº°Ò
R


AA R
+
=
2

¯Ë¹ºãºÎÒä º °˰m mÈ ¯ÈÏãÒÓ©² °È亰º¹¯«ÎËÓÓ©² º¹Ë¯Èº¯È
R
1
Ò
R
2
°¹ºãºÎÒËãÓ©äÒ °º°mËÓÓ©äÒ ÏÓÈËÓÒ«äÒ È}ÒË º

AA R
+
=
1
2


AA R
+
=
2
2
Ò

RRO
1
2
2
2
−=

~ÈäËÒäº
R
1
Ò
R
2
¹º¹º°¯ºËÓÒ°ä
°
ÒäËº°Ò°Ëä°º°
mËÓÓ©²mË}º¯ºmÈ¹ººäºÓÒ}ºääÒ¯ sººÈ°ºãȰӺ¹ °¹¯È
ãÒm©¯ÈmËÓ°mÈ

(

)(

)
RRRRRRRR RRRR O
1
2
2
2
1
2
12 21 2
2
1212
−=− + −= + =

jÏ ÓËm©¯ºÎËÓÓº°Ò Ò ãÒÓˮӺ°Ò
R
1
Ò
R
2
m °Òã ˺¯Ëä©  º¹Ë¯Èº¯

RR
12
+
È}ÎËÓËm©¯ºÎËÓÓ©®Ò¹ºªºä ÒÏ ¯ÈmËÓ°mÈ
(

)(

)
RRRR O
1212
−+=
°ãËË

RRO
12
−=
È}Òäº¯ÈϺä
R
°È亰º¹¯«ÎËÓÓ©®º¹Ë¯Èº¯º¹¯Ë
Ëã«Ëä©® ¹º
A
ºÓºÏÓÈÓº sº
QRA=
1
ÒÏÓÈÒ È}ÎË º¹¯ËËã«Ë°«
ºÓºÏÓÈÓº¹º
A

˺¯ËäÈº}ÈÏÈÓÈ
cÈÏËã
pm}ãÒºmº¹¯º°ˆ¯ÈÓ°ˆmº



                     , Ae
                  ( Ae   ) = ( A + Ae
                                      , e ) = λ ( e , e ) = λ δ ; i , j = [1, n]  sº ªˆº ºÏÓÈÈˈ ˆº
                      i    j           i  j     i i     j     i ij

                              1                       
                       ei′ =      Aei ; i = [1, n]        ˆÈ}ÎË­ÈÏÒ°Ò¹¯Òˆºäº¯ˆºÓº¯äÒ¯ºmÈÓÓ©®
                              λi                        
          
          °¯ÒäËä ÏÈ Ò°}ºä©® º¯ˆººÓÈã Ó©® º¹Ë¯Èˆº¯ Q   º¹Ë¯Èˆº¯ ¹Ë¯Ëmº«Ò®
               º¯ˆºÓº¯äÒ¯ºmÈÓÓ©® ­ÈÏÒ° {e1 , e2 ,..., en }  m º¯ˆºÓº¯äÒ¯ºmÈÓÓ©® ­ÈÏÒ°
                  {e1′ , e2′ ,..., en′ } Ò‚­ËÒä°«ˆºm}È˰ˆmË R äºÎÓºmÏ«ˆ º¹Ë¯Èˆº¯ Q −1 A 
          
                  iË®°ˆm҈Ëã Óº mº¹Ë¯m©² ÒäËˈ ä˰ˆº ¯ÈmËÓ°ˆmº A = QR
                                                                               {ºmˆº¯©² ÒÏ
                  °ººˆÓº ËÓÒ® R ei = Q −1 Ae
                                              = Q −1 λ e ′ = λ e ; i = [1, n]  °ãË‚ˈ ˆº ­ÈÏÒ°Ó©Ë
                                               i        i i     i i

                  ªãËäËӈ© ei , i = [1, n]  ˰ˆ  °º­°ˆmËÓÓ©Ë mË}ˆº¯© º¹Ë¯Èˆº¯È R  ºˆmËÈ ÒË
                  ¹ºãºÎ҈Ëã Ó©ä °º­°ˆmËÓÓ©ä ÏÓÈËÓÒ«ä                                   λi  È ÏÓÈ҈ äȈ¯ÒÈ R e  m ­È
                  ÏÒ°Ë {e1 , e2 ,..., en } ÒȺÓÈã Óȫҹºˆºä‚°Òääˈ¯Ò˰}È«‘ºÈm°Òã‚ãËää©
              º¹Ë¯Èˆº¯ R °È亰º¹¯«ÎËÓÓ©®
                 
                 
           ° º}ÈÎËä ÓÈ}ºÓË ËÒÓ°ˆmËÓÓº°ˆ  ¯ÈÏãºÎËÓÒ« {º mmËËÓÓ©² º­ºÏÓÈËÓÒ«²
                  °¹¯ÈmËãÒmº ¯ÈmËÓ°ˆmº A + A = R 2  ¹º°}ºã }‚ ÒÏ A = QR
                                                                                   Ò A + = R + Q +  °ãË‚ˈ
                  ˆº
                                                          A + A = R + Q + QR
                                                                                = R + Q −1QR
                                                                                                = R + R 
                  
                  ˆºm°Òã‚°È亰º¹¯«ÎËÓÓº°ˆÒ R  A + A = R 2 
                  
                  ¯Ë¹ºãºÎÒä ˆº °‚Ë°ˆm‚ ˆ mÈ ¯ÈÏãÒÓ©² °È亰º¹¯«ÎËÓÓ©² º¹Ë¯Èˆº¯È
                      R1  Ò R 2  ° ¹ºãºÎ҈Ëã Ó©äÒ °º­°ˆmËÓÓ©äÒ ÏÓÈËÓÒ«äÒ ˆÈ}ÒË ˆº A + A = R12 
                      A + A = R 2 Ò R 2 − R 2 = O 
                                 2        1       2
                  


                  ~ÈäˈÒ䈺 R1 Ò R 2 ¹º¹º°ˆ¯ºËÓÒ  °ä° ÒäË ˆº­‚ °Ò°ˆËä‚°º­°ˆ
                  mËÓÓ©²mË}ˆº¯ºmȹºˆºä‚ºÓÒ}ºä䂈ү‚ ˆsºˆºÈ°ºãȰӺ¹°¹¯È
                  mËãÒm©¯ÈmËÓ°ˆmÈ
                  
                                         R12 − R 22 = R12 − R1 R 2 + R 2 R1 − R 22 = ( R1 − R 2 )( R1 + R 2 ) = O 
                  
                  jÏ ÓËm©¯ºÎËÓÓº°ˆÒ Ò ãÒÓˮӺ°ˆÒ R1  Ò R 2  m °Òã‚ ˆËº¯Ëä©  º¹Ë¯Èˆº¯
                  R1 + R 2  ˆÈ}ÎË ÓËm©¯ºÎËÓÓ©® Ò ¹ºªˆºä‚ ÒÏ ¯ÈmËÓ°ˆmÈ ( R1 − R 2 )( R1 + R 2 ) = O 
                  °ãË‚ˈ R − R = O ‘È}Ò亭¯ÈϺä R °È亰º¹¯«ÎËÓÓ©®º¹Ë¯Èˆº¯º¹¯Ë
                                     1        2

                  Ëã«Ëä©® ¹º A  ºÓºÏÓÈÓº sº Q = R −1 A  Ò ÏÓÈ҈ ˆÈ}ÎË º¹¯ËËã«Ëˆ°«
           ºÓºÏÓÈÓº¹º A 
           
     ‘˺¯ËäȺ}ÈÏÈÓÈ