Лекции МГУ Артамонов Линал
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à㯯ë, ª®«ìæ ¨ ¯®«ï |
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¨áâ¥¬ë «¨¥©ëå ãà ¢¥¨© ¨ ¬ âà¨æë
1.¥â®¤ ãáá
áᬮâਬ ¯àאַ㣮«ìãî á¨á⥬㠫¨¥©ëå ãà ¢¥¨©
8:a:11: :x:1: : : :+: : : : : : |
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ª®íää¨æ¨¥âë ª®â®à®© aij; bj § ¤ ë. è 楫ì { ©â¨ ¢á¥ à¥è¥¨ï ¨ 㪠§ âì «£®à¨â¬ ¤«ï 宦¤¥¨ï ¢á¥å à¥è¥¨©.
¯à¥¤¥«¥¨¥ 1.1. ¥è¥¨¥¬ á¨á⥬ë (1) §ë¢ ¥âáï â ª®© ¡®à ç¨á¥« ( 1; : : : ; n), çâ® ¤«ï ¢á¥å i = 1; : : : ; m ¢ë¯®«¥ë à ¢¥áâ¢
ai1 1 + + ain n = bi:
ë ¡ã¤¥¬ à §«¨ç âì á«¥¤ãî騥 ¢¨¤ë á¨á⥬ (1).
¯à¥¤¥«¥¨¥ 1.2. ¨á⥬ (1) ¥á®¢¬¥áâ , ¥á«¨ ® ¥ ¨¬¥¥â à¥è¥¨ï. ¨á⥬ (1) ᮢ¬¥áâ , ¥á«¨ ® ¨¬¥¥â à¥è¥¨¥. ®¢¬¥áâ ï á¨á⥬ (1) ¥®¯à¥¤¥«¥ , ¥á«¨ ® ¨¬¥¥â ¡®«¥¥ ®¤®£® à¥è¥¨ï. ®¢¬¥áâ ï á¨á⥬ (1) ®¯à¥¤¥«¥ , ¥á«¨ ® ¨¬¥¥â ¥¤¨á⢥®¥ à¥è¥¨¥.
ë ¡ã¤¥¬ ᮢ¥àè âì àï¤ ¯à¥®¡à §®¢ ¨© á¨á⥬ë (1), ¥ ¬¥ïî騥 ¬®¦¥á⢠¥¥ à¥è¥¨©.
¯à¥¤¥«¥¨¥ 1.3. ¢¥ á¨áâ¥¬ë «¨¥©ëå ãà ¢¥¨© ¢¨¤ (1) íª¢¨¢ «¥âë, ¥á«¨ ®¨ ¨¬¥îâ ®¤¨ ª®¢ë¥ ¬®¦¥á⢠à¥è¥¨©.
ë ¡ã¤¥¬ ᮢ¥àè âì àï¤ ¯à®á⥩è¨å ¯à¥®¡à §®¢ ¨© á¨á⥬ë (1), á®åà ïîé¨å ¬®- ¦¥á⢠à¥è¥¨©. ¬¥â¨¬, çâ® ¢á¥ ¨ä®à¬ æ¨ï ® á¨á⥬¥ (1) ᮤ¥à¦¨âáï ¢ ¥¥ â ¡«¨æ¥ ¥¥
ª®íää¨æ¨¥â®¢.
¯à¥¤¥«¥¨¥ 1.4. âà¨æ¥© á¨á⥬ë (1) §ë¢ ¥âáï ¯àאַ㣮«ì ï â ¡«¨æ
01
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áè¨à¥®© ¬ âà¨æ¥© á¨á⥬ë (1) §ë¢ ¥âáï ¯àאַ㣮«ì ï â ¡«¨æ
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¬¥ç ¨¥ 1.5. ®£¤ à áè¨à¥ãî ¬ âà¨æã á¨á⥬ë (1) ®¡®§ ç îâ ç¥à¥§
0 a11 |
a1n |
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amn |
bm A |
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6 1.
¯à¥¤¥«¥¨¥ 1.6. «¥¤ãî騥 ¯à¥®¡à §®¢ ¨ï á¨á⥬ë (1) (¥¥ (à áè¨à¥®©) ¬ â- à¨æë) §ë¢ îâáï í«¥¬¥â à묨:
} ¯à¨¡ ¢«¥¨¥ ª ®¤®¬ã ãà ¢¥¨î (áâப¥) ¤à㣮£® ãà ¢¥¨ï (¤à㣮© áâப¨),
㬮¦¥®£®(®©) ¯à®¨§¢®«ì®¥ ç¨á«®; ~ 㬮¦¥¨¥ ãà ¢¥¨¥ (áâப¨) ¥ã«¥¢®¥ ç¨á«®.
¥®à¥¬ 1.7. à¨ í«¥¬¥â àëå ¯à¥®¡à §®¢ ¨ïå ¯¥à¥å®¤¨¬ ª íª¢¨¢ «¥â®© á¨á- ⥬¥.
®ª § ⥫ìá⢮. ।¯®«®¦¨¬, çâ® ¬ë ᮢ¥àè ¥¬ ¯à¥®¡à §®¢ ¨¥ ⨯ }, ¨¬¥®, ª i-®¬ã ãà ¢¥¨î ¯à¨¡ ¢«ï¥¬ j-®¥, 㬮¦¥®¥ . ᫨ ( 1; : : : ; n) { à¥è¥¨¥ ¨á室- ®© á¨á⥬ë (1). ᥠãà ¢¥¨ï ®¢®© á¨á⥬ë, ªà®¬¥ i-£®, ¥ ¨§¬¥¨«¨áì. ᫨ ¬ë ¯®¤áâ ¢¨¬ ¡®à ( 1; : : : ; n) ¢ i-®¥ ãà ¢¥¨¥ ®¢®© á¨á⥬ë, â® ¯®«ã稬
(ai1 + aj1) 1 + + (ain + ajn) n =
(ai1 1 + + ain n) + (aj1 1 + + ajn n) = bi + + bj:
ª¨¬ ®¡à §®¬, ( 1; : : : ; n) ï¥âáï à¥è¥¨¥¬ ®¢®© á¨á⥬ë. ®áª®«ìªã ¨á室 ï á¨á- ⥬ë (1) ¯®«ãç ¥âáï ¨§ ®¢®© á¨á⥬ë í«¥¬¥â àë¬ ¯à¥®¡à §®¢ ¨¥¬ ¯à¨¡ «¥¨¥¬ ª i-®¬ã ãà ¢¥¨î j-£®, 㬮¦¥®£® , â® «®£¨ç®, ª ¦¤®¥ à¥è¥¨¥ ®¢®© á¨á⥬ë
ï¥âáï à¥è¥¨¥¬ ¨á室®© á¨á⥬ë.
¯à ¦¥¨¥ 1.8. ®ª § âì, ç⮠ᮢ¥àè ï í«¥¬¥â àë¥ ¯à¥®¡à §®¢ ¨ï á® áâப ¬¨ ¬ âà¨æë ¬®¦® ¢ ¥© ¯¥à¥áâ ¢¨âì «î¡ë¥ ¤¢¥ áâப¨.
㤥¬ ¯à¨¢®¤¨âì ¬ âà¨æã á¨áâ¥¬ë ª ¨¡®«¥¥ ¯à®á⮬ã { áâ㯥ç ⮬㠢¨¤ã.¯à¥¤¥«¥¨¥ 1.9. âà¨æ (3) §ë¢ ¥âáï áâ㯥ç ⮩, ¥á«¨
(1)¨¦¥ ã«¥¢®© áâப¨ à ᯮ«®¦¥ë ⮫쪮 ã«¥¢ë¥ áâப¨;
(2)¯¥à¢ë© ¥ã«¥¢®© ª ¦¤®© áâப¨ à ¢¥ 1;
(3) ¥á«¨ ¯¥à¢ë© ¥ã«¥¢®© i-®© áâப¨ à ᯮ«®¦¥ ¬¥á⥠(i; ki), â®
(a)ki+1 > ki;
(b)¢á¥ í«¥¬¥âë aj;ki = 0 ¤«ï ¢á¥å j 6= i.
¥®à¥¬ 1.10. ¦¤ ï ¬ âà¨æ ª®¥çë¬ ç¨á«®¬ í«¥¬¥â àëå ¯à¥®¡à §®¢ ¨© áâப ¯à¨¢®¤¨âáï ª áâ㯥ç ⮬㠢¨¤ã.
®ª § ⥫ìá⢮. ãáâì ¬ âà¨æ A ¨¬¥¥â ¢¨¤ (3). ᫨ A = 0, â® ® 㦥 ¨¬¥¥â
áâ㯥ç âë© ¢¨¤.
ãáâì A 6= 0. 㤥¬ ¢¥á⨠¤®ª § ⥫ìá⢮ ¨¤ãªæ¨¥© ¯® ç¨á«ã áâப m. ¥§ ®£à ¨ç¥-
¨ï ®¡é®á⨠¬®¦® áç¨â âì, çâ® ¢ ¯¥à¢®¬ á⮫¡æ¥ ¥áâì ¥ã«¥¢®© í«¥¬¥â ai1. ᫨ i = 1, ⮠㬮¦¨¬ 1-ãî áâபã a111. â ª, ¬®¦® ¯à¥¤¯®« £ âì, çâ® a11 = 1. «¥¤®¢ ⥫ì®,
¥á«¨ m = 1, ⮠⥮६ ¤®ª § .
ãáâì m > 1, ¨ ¤«ï m 1 ⥮६ ¤®ª § . «ï ª ¦¤®£® i > 1 ¢ëç⥬ ¨§ i-®© áâப¨ ¯¥à¢ãî áâபã, 㬮¦¥ãî ai1a111. ®¢®© ¬ âà¨æ¥ ¢á¥ ª®íää¨æ¨¥âë ai1 = 0; i > 1.
бᬮва¨¬ ¯®¤¬ ва¨жг B ¢ A, ¯®«гз ойгобп ®в¡а бл¢ ¨¥¬ ¯¥а¢®© бва®ª¨. ® ¨¤гªж¨¨ ¬®¦® бз¨в вм, зв® ¬ ва¨ж B ¨¬¥¥в бвг¯¥з вл© ¢¨¤. гбвм ¢ ¬ ва¨ж¥ B ¯¥а¢л¥ ¥г«¥¢л¥ н«¥¬¥вл а б¯®«®¦¥л ¢ бв®«¡ж е б ®¬¥а ¬¨ 1 < k2 < k3 < .ëç⥬ ¨§ ¯¥à¢®© áâப¨ 2-ãî áâபã, 㬮¦¥ãî a1;k2 , âà¥âìî áâபã 3-ãî áâபã, 㬮¦¥ãî a1;k3 , ¨ â. ¤.
¯à¥¤¥«¥¨¥ 1.11. ãáâì ¬ âà¨æ á¨á⥬ë (1) ¨¬¥¥â áâ㯥ç âë© ¢¨¤. §®¢¥¬ ¥¨§¢¥áâãî xi £« ¢®©, ¥á«¨ ¢ ¥ª®â®à®¬ ãà ¢¥¨¨ ¢á¥ ª®íää¨æ¨¥âë ¯à¨ x1; : : : ; xi 1 à ¢ë ã«ï, ª®íää¨æ¨¥â ¯à¨ xi ®â«¨ç¥ ®â ã«ï (¨ ¯®â®¬ã à ¢¥ 1). ¢á¥ ®áâ «ìë¥ ¥¨§¢¥áâë¥ §®¢¥¬ ᢮¡®¤ë¬¨.
2. |
7 |
ਬ¥¨¬ ⥮६ë 1.7, 1.10 ª ¨áá«¥¤®¢ ¨î á¨á⥬ë (1). ᨫã 㪠§ ëå ⥮६ ¬®¦® áç¨â âì, çâ® à áè¨à¥ ï ¬ âà¨æ á¨á⥬ë (1) ¨¬¥¥â áâ㯥ç âë© ¢¨¤.
ãáâì ¥¥ ¯®á«¥¤ïï ¥ã«¥¢ ï áâப ¨¬¥¥â ¢¨¤
(0; : : : ; 0; 1):
â® ®§ ç ¥â, çâ® á¨á⥬ë (1) ᮤ¥à¦¨â ãà ¢¥¨¥
0x1 + + 0xn = 1;
çâ® ¥¢®§¬®¦®. «¥¤®¢ ⥫ì®, ¢ í⮬ á«ãç ¥ á¨á⥬ |
¥á®¢¬¥áâ . |
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ãáâì ¢ A ¥â áâப¨ (4). |
।¯®«®¦¨¬ ¤«ï ¯à®áâ®âë, çâ® ¯¥à¥¬¥ë¥ |
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£« ¢ë¥, xr+1; : : : ; xn ᢮¡®¤ë¥. ®£¤ á¨á⥬ |
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¥à¥®áï ᢮¡®¤ë¥ ¯¥à¥¬¥ë¥ ¢ ¯à ¢ãî ç áâì, ¯®«ãç ¥¬ ¢ëà ¦¥¨¥ £« ¢ëå ¥¨§¢¥áâ- ëå ç¥à¥§ ᢮¡®¤ë¥
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a1;r+1xr+1 |
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ï ᢮¡®¤ë¬ ¥¨§¢¥áâë¬ ¯à®¨§¢®«ìë¥ § 票ï, ¬ë ®¤®§ ç® |
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室¨¬ § ç¥¨ï £« ¢ëå ¥¨§¢¥áâëå. â ª, á¨á⥬ |
ᮢ¬¥áâ , ¨, ¥á«¨ ¥áâì ᢮¡®¤ë¥ |
¥¨§¢¥áâë¥, â® á¨á⥬ ¥®¯à¥¤¥«¥ . ᫨ ¢á¥ ¥¨§¢¥áâë¥ £« ¢ë¥, â® á¨á⥬ ®¯à¥- ¤¥«¥ .
¯à¥¤¥«¥¨¥ 1.12. ¨á⥬ (1) ®¤®à®¤ , ¥á«¨ ¢á¥ ¥¥ ᢮¡®¤ë¥ ç«¥ë ã«¥¢ë¥, â. ¥. b1 = = bm = 0.
।«®¦¥¨¥ 1.13. ᫨ ¢ ®¤®à®¤®© á¨á⥬¥ ç¨á«® ¥¨§¢¥áâëå n ¡®«ìè¥ ç¨á« ãà ¢¥¨© m, â® á¨á⥬ ¥®¯à¥¤¥«¥ .
®ª § ⥫ìá⢮. ਢ¥¤¥¬ á¨á⥬㠪 áâ㯥ç ⮬㠢¨¤ã. á®, ç⮠ᮢ ¯®«ã稬 ®¤®à®¤ãî á¨á⥬ã, ¯à¨ç¥¬ ç¨á«® £« ¢ëå ¥¨§¢¥áâëå ¥ ¯à¥¢®á室¨â ç¨á« ¥ã«¥¢ëå ãà ¢¥¨©, â. ¥. ¥ ¢á¥ ¥¨§¢¥áâë¥ £« ¢ë¥.
2. âà¨æë ¨ ®¯¥à 樨 ¤ ¨¬¨
¯à¥¤¥«¥¨¥ 1.14. Mat(n m) { ¢á¥å ¬ âà¨æ (¯àאַ㣮«ìëå â ¡«¨æ) á n áâப ¬¨ ¨ m á⮫¡æ ¬¨. ᫨ A 2 Mat(n m), â® ¬ë ¡ã¤¥¬ â ª¦¥ ¯¨á âì A = An m: ᫨ An m = (aij); Bn m = (bij), â® ¯®« £ ¥¬ A + B = (aij + bij). ஬¥ ⮣®, An m = ( aij).
।«®¦¥¨¥ 1.15. ãáâì A; B; C 2 Mat(n m) ¨ ; { ç¨á« . ®£¤ á¯à ¢¥¤«¨¢ë á«¥¤ãî騥 8 ªá¨®¬ ¢¥ªâ®à®£® ¯à®áâà á⢠:
(1)A + B = B + A;
(2)A + (B + C) = (A + B) + C;
(3) |
¥á«¨ 0 { ã«¥¢ ï ¬ âà¨æ (¢á¥ ¥¥ ª®íää¨æ¨¥âë à ¢ë ã«î), â® A + 0 = A ¤«ï |
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«î¡®© ¬ âà¨æë A; |
(4) |
¤«ï «î¡®© ¬ âà¨æë A áãé¥áâ¢ã¥â â ª ï ¬ âà¨æ A, çâ® A + ( A) = 0; |
(5)(A + B) = A + B;
(6)( + )A = A + A;
(7)( )A = ( A);
(8)1A = A.
8 1.
®ª § ⥫ìá⢮. ਢ¥¤¥¬, ¯à¨¬¥à, ¤®ª § ⥫ìá⢮ ¯¥à¢®£® ã⢥ত¥¨ï, ᫨ A = (aij); B = (bij), â® A + B = (aij + bij) = (bij + aij) = B + A. áâ «ìë¥ ã⢥ত¥¨ï ¤®ª §ë¢ îâáï «®£¨ç®î
¯à¥¤¥«¥¨¥ 1.16. ãáâì
An m = (aij); Cm k = (cst): |
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®£¤ D = AC 2 Mat(n k) = (dis), £¤¥ ¤«ï ¢á¥å i = 1; : : : ; n; |
s = 1; : : : ; k |
dis = ai1d1s + + aindns |
(7) |
।«®¦¥¨¥ 1.17. ¬®¦¥¨¥ ¬ âà¨æ áá®æ¨ ⨢®, â.¥. (AC)F = A(CF ) ¤«ï
«î¡ëå ¬ âà¨æ
A 2 Mat(n m); C 2 Mat(m k); F 2 Mat(k l):
®ª § ⥫ìá⢮. ãáâì
A = An m = (aij); C = Cm k = (cst); F = Fk l = (ftq):
᫨ D ¨§ ®¯à¥¤¥«¥¨ï 1.16, â® ¯® (7) ¬¥á⥠(i; q) ¢ ¬ âà¨æ¥ (AC)F = DF á⮨â í«¥¬¥â
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CF = U = (ui;) 2 Mat(m l);
â® ¬¥á⥠(i; q) ¢ ¬ âà¨æ¥ A(CF ) = AU á⮨â í«¥¬¥â
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ai;u;q = |
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§ (8), (9) ¢ë⥪ ¥â ã⢥ত¥¨¥. |
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।«®¦¥¨¥ 1.18. ¯à ¢¥¤«¨¢ë à ¢¥á⢠: |
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(AB) = ( A)B = A( B): |
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(2) |
A(B + C) = AB + AC; (A + U)V = AV + UV: |
®ª § ⥫ìá⢮. |
®ª ¦¥¬, ¯à¨¬¥à, ¢â®à®¥ ã⢥ত¥¨¥. ãáâì |
(bij); C = (cij). ®£¤ |
¬¥á⥠(i; j) ¢ ¬ âà¨æ¥ A(B + C) áâநâ í«¥¬¥â |
(9)
A = (aij); B =
ª®â®àë© à ¢¥ í«¥¬¥âã,X |
X |
X |
AB + AC |
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aik(bkj + ckj) = |
aikbkj + |
aikckj; |
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áâ®ï饬ã â® ¦¥ ¬¥á⥠¢ ¬ âà¨æ¥ |
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¬ ва¨ж A(B+C) ¨ AB+AC б®¢¯ ¤ ов, в® ®¨ а ¢л. «®£¨з® ¯а®¢¥аповбп ®бв «мл¥ гв¢¥а¦¤¥¨п.
¯à¥¤¥«¥¨¥ 1.19. ãáâì A = (aij) 2 Mat(n). «¥¤®¬ tr A §ë¢ ¥âáï a11 + +ann.
।«®¦¥¨¥ 1.20. ãáâì A; B 2 Mat(n). ®£¤ tr(AB) = tr(BA).
®ª § ⥫ìá⢮. ãáâì A = (aij); B = (bij): ®£¤ ¬¥á⥠(i; i) ¢ ¬ âà¨æ¥ AB
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á⮨â Pj=1 aijbji, ®âªã¤ |
tr(AB) = |
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aijbji: |
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i;j=1 |
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«®£¨ç®, |
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tr(BA) = |
bstats = |
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atsbst = tr(AB): |
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s;t=1 |
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¯à¥¤¥«¥¨¥ 1.21. ¨¬¢®« ஥ª¥à ij à ¢¥ 1, ¥á«¨ i = j, ¨ 0, ¥á«¨ i 6= j. ¤¨- ¨ç ï ¬ âà¨æ E = En 2 Mat(n) { íâ® ¬ âà¨æ , ¢ ª®â®à®© ¬¥á⥠(i; j) á⮨â ᨬ¢®«à®¥ª¥à ij.
।«®¦¥¨¥ 1.22. ãáâì A 2 Mat(n m): ®£¤ EnA = A = AEm.
®ª § ⥫ìá⢮. ãáâì A = (aij). ®£¤ ¬¥á⥠(i; j) ¢ ¬ âà¨æ¥ EnA á⮨â
n
X
ikakj = iiaij = aij;
k=1
â. ¥. EnA = A:
¯à¥¤¥«¥¨¥ 1.23. ãáâì A 2 Mat(n m). ®£¤ âà ᯮ¨à®¢ ï ¬ âà¨æ tA =
A 2 Mat(m n) { íâ® ¬ âà¨æ , ¢ ª®â®à®© |
¬¥á⥠(i; j) á⮨â í«¥¬¥â aji ¬ âà¨æë A. |
।«®¦¥¨¥ 1.24. t(A + B) =tA +tB; |
t( A) = tA; t(AC) =tCtA: |
®ª § ⥫ìá⢮. ®ª ¦¥¬, ¯à¨¬¥à, ¯®á«¥¤¥¥ ã⢥ত¥¨¥. ¬ âà¨æ¥ t(AC)
PP
¬¥á⥠(i; j) á⮨â k ajkcki = |
k ckiajk, â. ¥. í«¥¬¥â, áâ®ï饩 ⮬ ¦¥ ¬¥á⥠¢ ¬ âà¨æ¥ |
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«®£¨ç® ¤®ª §ë¢ îâáï ®áâ «ìë¥ ã⢥ত¥¨ï. |
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¯à¥¤¥«¥¨¥ 1.25. âà¨çë¥ ¥¤¨¨æë Eij |
2 Mat(n m) { íâ® ¬ âà¨æë Eij, ¢ |
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ª®â®àëå ¬¥á⥠(s; t) á⮨â í«¥¬¥â si tj, â. ¥. |
¬¥á⥠(i; j) á⮨â 1, ¨ ¢á¥ ®áâ «ìë¥ |
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í«¥¬¥âë à ¢ë 0. |
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¯à ¦¥¨¥ 1.26. ®ª § âì, çâ® |
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।«®¦¥¨¥ 1.27. E Ers = jrEis:
®ª § ⥫ìá⢮. ¬¥á⥠(u; v) ¢ EijErs á⮨â í«¥¬¥â
(
X( ui pj)( pi vs) = 1; u = i = p = j; v = s; 0 ¢ ¯à®â¨¢®¬ á«ãç ¥.
p
âáî¤ ¢ë⥪ ¥â ã⢥ত¥¨¥.
«¥¤á⢨¥ 1.28. ãáâì A = (ars) 2 Mat(n m). ®£¤
EijA = aj1Ei1 + + ajmEim;
AEij = a1iE1j + + anjEni:
¥®à¥¬ 1.29. â®¡ë ¢ ¬ âà¨æ¥ A 2 Mat(n m) ª i-®© áâப¥ ¯à¨¡ ¢¨âì j-ãî, 㬮¦¥ãî 㦮 à áᬮâà¥âì ¬ âà¨æã (En + Eij)A.
®ª § ⥫ìá⢮. ® ¯à¥¤«®¦¥¨ï¬ 1.18, 1.27, ã¯à ¦¥¨ï¬ 1.22, | ¨ á«¥¤á⢨î 1.28
(En + Eij)A = EnA + EijA =
P
A + (aj1Ei1 + + ajmEim) = rs arsErs + ( aj1)Ei1 + + ( ajm)Eim =
PP
r6=i;s arsErs + i;s(ais + ajs)Eis:
«¥¤á⢨¥ 1.30. â®¡ë ¢ ¬ âà¨æ¥ A 2 Mat(n m) ª i-®¬ã á⮫¡æ㠯ਡ ¢¨âì j-ë©, 㬮¦¥ãî 㦮 à áᬮâà¥âì ¬ âà¨æã A(Em + Eji).
¡®§ 票¥ 1.31. ®«®¦¨¬ Di( ) = En + ( 1)Eii 2 Mat(n).
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¥®à¥¬ 1.32. â®¡ë ¢ ¬ âà¨æ¥ A 2 Mat(n m) i-ãî áâபã (á⮫¡¥æ) 㬮¦¨âì 㦮 à áᬮâà¥âì ¬ âà¨æã Di( )A (ADi( )).
®ª § ⥫ìá⢮. ãáâì A = (aij). |
¬¥â¨¬, çâ® ¢ ¬ âà¨æ¥ Di( ) ¬¥á⥠(s; t) |
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á⮨â st + ( 1) is it. ®í⮬㠢 ¬ âà¨æ¥ Di( )A |
¬¥á⥠(p; q) á⮨â |
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r( pr + ( 1) ip ir)arq = |
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+ ( 1) r ip irarq = |
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¬¥ç ¨¥ 1.33. â¥à¬¨ å ¬ âà¨ç®£® 㬮¦¥¨ï 㤮¡® § ¯¨áë¢ âì á¨áâ¥¬ë «¨- ¥©ëå ãà ¢¥¨©. ¬¥®, á¨á⥬ë (1) ¨¬¥¥â ¢¨¤ AX = b, £¤¥ A { ¬ âà¨æ (2) á¨á⥬ë
(1),
0 1 x1
X = B ... C @ A
xn
{ á⮫¡¥æ ¥¨§¢¥áâëå,
0 1
b1
b = B ... C @ A
bm
{ á⮫¡¥æ ᢮¡®¤ëå ç«¥®¢.