L-10-Spring2018 (Грешнов - Лекции (1-15))

„ â : 10.04.2018‹…Š–ˆŸ ü10Š®­¨ç¥áª¨¥ á¥ç¥­¨ï. ‡ ä¨ªá¨à㥬 ¢ ¯à®áâà ­á⢥ ­¥ª®â®àãî ¯àï¬ãî l, ¢ë¡¥à¥¬­  ­¥© ¯à®¨§¢®«ì­ë¬ ®¡à §®¬ â®çªã O ¨ ¯à®¢¥¤¥¬ ç¥à¥§ ­¥¥ ­¥ª®â®àãî ¯àï¬ãî hâ ªãî, ç⮠㣮« α, ®¡à §®¢ ­­ë© ¯àï¬ë¬¨ l ¨ h, ®áâàë©. ã¤¥¬ ¢à é âì ¯àï¬ãîh ¢®ªà㣠¯àאַ© l â ª, ç⮡ë 㣮« ¬¥¦¤ã ¯àï¬ë¬¨ ¢á¥ ¢à¥¬ï ®áâ ¢ «áï à ¢­ë¬α. ’ ª¨¬ ®¡à §®¬, ¬ë ¯®«ã稬 ªà㣮¢®© ª®­ãá Kl,α á ®áìî l ¨ 㣫®¬ ¢à é¥­¨ï α.Ž¯à¥¤¥«¥­¨¥ 10.1. Š®­¨ç¥áª¨¬ á¥ç¥­¨¥¬ ¨«¨ ª®­¨ª®© ­ §ë¢ ¥âáï ªà¨¢ ï, ¯®ª®â®à®© ¯¥à¥á¥ª ¥â ªà㣮¢®© ª®­ãá ¯à®¨§¢®«ì­ ï ¯«®áª®áâì, ­¥ ¯à®å®¤ïé ï ç¥à¥§¥£® ¢¥à設ã.’¥®à¥¬  10.1 (® ª®­¨ç¥áª¨å á¥ç¥­¨ïå). ‹î¡ ï ª®­¨ª , ªà®¬¥ ®ªà㦭®áâ¨, |í««¨¯á, £¨¯¥à¡®«  ¨«¨ ¯ à ¡®« .„®ª § â¥«ìá⢮. ãáâì γ | ªà¨¢ ï, ¯® ª®â®à®© ¯«®áª®áâì σ ¯¥à¥á¥ª ¥â ª®­ãáKl,α . ‚¯¨è¥¬ ¢ ª®­ãá Kl,α áä¥àã (áä¥à  „ ­¤¥«¥­ ), ª®â®à ï ª á ¥âáï ª®­ãá ¯® ®ªà㦭®á⨠(業âà â ª®© áä¥àë ¢á¥£¤  «¥¦¨â ­  ¯àאַ© l),   â ª¦¥ ª á ¥âáﯫ®áª®áâ¨ σ ¢ â®çª¥ F (®â¬¥â¨¬, çâ® ¢ § ¢¨á¨¬®á⨠®â ¢§ ¨¬­®£® à á¯®«®¦¥­¨ïσ ¨ Kl,α áä¥à  „ ­¤¥«¥­  ¬®¦¥â ¡ëâì ¢ë¡à ­  ­¥ ¢á¥£¤  ¥¤¨­á⢥­­ë¬ ®¡à §®¬).ãáâì ω | ¯«®áª®áâì, ¢ ª®â®à®© «¥¦¨â ®ªà㦭®áâì ª á ­¨ï áä¥àë á ª®­ãᮬ.‚롥६ ¯à®¨§¢®«ì­® â®çªã M ∈ γ .

à®¢¥¤¥¬ ç¥à¥§ ¢¥à設㠪®­ãá  ¨ â®çªãM ¯àï¬ãî ¨ ®¡®§­ ç¨¬ ç¥à¥§ B â®çªã ¯¥à¥á¥ç¥­¨ï í⮩ ¯àאַ© á ¯«®áª®áâìî w.‡ ¬¥â¨¬, çâ®d(B, M ) = d(F, M )(10.1)(¯®áª®«ìªã B, F | â®çª¨ ª á ­¨ï áä¥àë ¯àï¬ë¬¨, ¯à®å®¤ïé¨å ç¥à¥§ ®¡éãî â®çªã M ). ãáâì δ = σ ∩ ω (íâ® ¯¥à¥á¥ç¥­¨¥ ­¥ ¯ãáâ®, ¯®áª®«ìªã ¬ë à áᬠâਢ ¥¬ª®­¨ªã, ®â«¨ç­ãî ®â ®ªà㦭®áâ¨). Ž¯ãá⨬ ¨§ â®çª¨ M ¯¥à¯¥­¤¨ªã«ïà ­  ¯àï¬ãî δ , ¨ ¯ãáâì A ∈ δ | ª®­¥æ í⮣® ¯¥à¯¥­¤¨ªã«ïà . ãáâì α | 㣮« ¬¥¦¤ã¯«®áª®áâﬨ σ ¨ ω, ⮣¤ d(M, ω )sin α =.(10.2)d(A, M )ãáâì β | 㣮« ¬¥¦¤ã ¯«®áª®áâìî ω ¨ ¯àאַ©, ¯à®å®¤ï饩 ç¥à¥§ ¢¥à設㠪®­ãá ¨ â®çªã M , ⮣¤ d(M, ω )sin β =.(10.3)d(M, B )“ç¨â뢠ï (10.1){(10.3), ¬ë ¯®«ãç ¥¬d(F, M )d(A, M )sin α=.sin β12®­ïâ­®, çâ® ¢¥«¨ç¨­ë 㣫®¢ α, β ­¥ § ¢¨áïâ ®â ¢ë¡®à  â®çª¨ M ∈ γ ,   íâ® §­ ç¨â,çâ® ¢ ¯«®áª®á⨠σ ¬ë ­ è«¨ â®çªã F ¨ ¯àï¬ãî δ â ª¨¥, çâ® ¤«ï ¯à®¨§¢®«ì­®©F,M )â®çª¨ M ∈ γ ⊂ σ ®â­®è¥­¨¥ dd((A,M) à ¢­® ¯®áâ®ï­­®© ¢¥«¨ç¨­¥.

’®£¤  ⥮६  10.1¢ë⥪ ¥â ¨§ ⥮६ë 3.4 «¥ªæ¨¨ ü3.¥22290 Œ­¨¬ë¥ ª®­ãáë ®¯à¥¤¥«ïîâáï ãà ¢­¥­¨¥¬ xa2 + yb2 + zc2 = 0.…¤¨­á⢥­­ ï ¢¥é¥á⢥­­ ï â®çª , ¯à¨­ ¤«¥¦ é ï ¬­¨¬®¬ã ª®­ãáã | ­ ç «®ª®®à¤¨­ â (0, 0, 0).222100 ««¨¯á®¨¤ë ®¯à¥¤¥«ïîâáï ãà ¢­¥­¨¥¬ xa2 + yb2 + zc2 = 1. Š®®à¤¨­ â­ë¥¯«®áª®á⨠| ¯«®áª®á⨠ᨬ¬¥âਨ í««¨¯á®¨¤ , ­ ç «® ª®®à¤¨­ â | 業âà ᨬ¬¥âਨ í««¨¯á®¨¤ . …᫨ a = b, â® í««¨¯á®¨¤ ¯®«ãç ¥âáï ¢à é¥­¨¥¬ í««¨¯á x2z2a2 + c2 = 1, x = 0, ¢®ªà㣠®á¨ OZ . …᫨ ¢ ãà ¢­¥­¨¨ í««¨¯á®¨¤  a = b = c, â® ¬ë¯®«ãç ¥¬ ãà ¢­¥­¨¥ áä¥àë à ¤¨ãá  a.110 Œ­¨¬ë¥ í««¨¯á®¨¤ë ®¯à¥¤¥«ïîâáï ãà ¢­¥­¨¥¬x2a222+ yb2 + zc2 = −1.222120 Ž¤­®¯®«®áâ­ë¥ £¨¯¥à¡®«®¨¤ë ®¯à¥¤¥«ïîâáï ãà ¢­¥­¨¥¬ xa2 + yb2 − zc2 = 1.««¨¯á( 2y2xa2 + b 2 = 1 ,z = 0,­ §ë¢ ¥âáï £®à«®¢ë¬ í««¨¯á®¬ ®¤­®¯®«®áâ­®£® £¨¯¥à¡®«®¨¤ .

Š®®à¤¨­ â­ë¥ ¯«®áª®á⨠| ¯«®áª®á⨠ᨬ¬¥âਨ ®¤­®¯®«®áâ­®£® £¨¯¥à¡®«®¨¤ , ­ ç «® ª®®à¤¨­ â |業âà ᨬ¬¥âਨ ®¤­®¯®«®áâ­®£® £¨¯¥à¡®«®¨¤ . …᫨ a = b, â® ®¤­®¯®«®áâ­ë©£¨¯¥à¡®«®¨¤ ¯®«ãç ¥âáï ¢à é¥­¨¥¬ £¨¯¥à¡®«ë( 2x2= 1,− zc2y = 0,a2¢®ªà㣠®á¨ OZ .222130 „¢ã¯®«®áâ­ë¥ £¨¯¥à¡®«®¨¤ë ®¯à¥¤¥«ïîâáï ãà ¢­¥­¨¥¬ − xa2 − yb2 + zc2 =1. Š®®à¤¨­ â­ë¥ ¯«®áª®á⨠| ¯«®áª®á⨠ᨬ¬¥âਨ ¤¢ã¯®«®áâ­®£® £¨¯¥à¡®«®¨¤ ,­ ç «® ª®®à¤¨­ â | 業âà ᨬ¬¥âਨ ¤¢ã¯®«®áâ­®£® £¨¯¥à¡®«®¨¤ . …᫨ a = b,â® ¤¢ã¯®«®áâ­ë© £¨¯¥à¡®«®¨¤ ¯®«ãç ¥âáï ¢à é¥­¨¥¬ £¨¯¥à¡®«ë(¢®ªà㣠®á¨ OZ .22+ zc2 = 1,y = 0,− xa222140 ««¨¯â¨ç¥áª¨¥ ¯ à ¡®«®¨¤ë ®¯à¥¤¥«ïîâáï ãà ¢­¥­¨¥¬ 2z = xp + yq ,p, q > 0.

à¨ ª ¦¤®¬ 䨪á¨à®¢ ­­®¬ z = z0 > 0 ¬ë ¯®«ãç ¥¬ ãà ¢­¥­¨¥ í««¨¯á 3x22+ 2yz0 q = 1. Š®®à¤¨­ â­ë¥ ¯«®áª®á⨠ZOX , ZOY | ¯«®áª®á⨠ᨬ¬¥âਨ í««¨¯â¨ç¥áª®£® ¯ à ¡®«®¨¤ . …᫨ p = q, â® í««¨¯â¨ç¥áª¨© ¯ à ¡®«®¨¤ ¯®«ãç ¥âáï¢à é¥­¨¥¬ ¯ à ¡®«ë(2z = y2q ,(10.4)x = 0,¢®ªà㣠®á¨ OZ (¯ à ¡®«®¨¤ á ¯ à ¬¥â஬ p). à¨ ¯®¤áâ ­®¢ª¥ §­ ç¥­¨ï x = x0 ¢ãà ¢­¥­¨¥ í««¨¯â¨ç¥áª®£® ¯ à ¡®«®¨¤  ¬ë ¯®«ã稬 ãà ¢­¥­¨¥ ¯ à ¡®«ë2z0 pz=x202p+y2,2q³2´«¥¦ é¥© ¢ ¯«®áª®á⨠x = x0 , á ¢¥à設®© ¢ â®çª¥ x0 , 0, x2p0 .

’ ª¨¬ ®¡à §®¬, í««¨¯â¨ç¥áª¨© ¯ à ¡®«®¨¤ ¯®«ãç ¥âáï ý¯ à ««¥«ì­ë¬þᤢ¨£®¬ ý¯®¤¢¨¦­®©þ ¯ à (x2z = 2p ,¡®«ë (10.4) ¢¤®«ì ý­¥¯®¤¢¨¦­®©þ ¯ à ¡®«ë.y=022150 ƒ¨¯¥à¡®«¨ç¥áª¨¥ ¯ à ¡®«®¨¤ë ®¯à¥¤¥«ïîâáï ãà ¢­¥­¨¥¬ 2z = xp − yq ,p, q > 0. Š®®à¤¨­ â­ë¥ ¯«®áª®á⨠ZOX , ZOY | ¯«®áª®á⨠ᨬ¬¥âਨ £¨¯¥à¡®«¨ç¥áª®£® ¯ à ¡®«®¨¤ .

à¨ ¯®¤áâ ­®¢ª¥ ª ¦¤®£® §­ ç¥­¨ï z0 6= 0 ¢ ãà ¢­¥­¨¥£¨¯¥à¡®«¨ç¥áª®£® ¯ à ¡®«®¨¤  ¬ë ¯®«ãç ¥¬ ãà ¢­¥­¨¥ £¨¯¥à¡®«ë(2x22z0 p− 2yz0 qz = z0 ,= 1,¯à¨ ¯®¤áâ ­®¢ª¥ §­ ç¥­¨ï z0 = 0 | ãà ¢­¥­¨¥ ¯àï¬ëå( 2x2= yq ,z = 0.pà¨ ¯®¤áâ ­®¢ª¥ §­ ç¥­¨ï y = 0 ¢ ãà ¢­¥­¨¥ £¨¯¥à¡®«¨ç¥áª®£® ¯ à ¡®«®¨¤  ¬ë¯®«ãç ¥¬ ãà ¢­¥­¨¥ ¯ à ¡®«ë(2z = x2p ,(10.5)y = 0.à¨ ¯®¤áâ ­®¢ª¥ §­ ç¥­¨ï x = x0 ¢ ãà ¢­¥­¨¥ £¨¯¥à¡®«¨ç¥áª®£® ¯ à ¡®«®¨¤  ¬ë¯®«ãç ¥¬ ãà ¢­¥­¨¥ ¯ à ¡®«ëz=−y22q+x20,2p(10.6)³2´«¥¦ é¥© ¢ ¯«®áª®á⨠x = x0 , á ¢¥à設®© ¢ â®çª¥ x0 , 0, x2p0 . ’ ª¨¬ ®¡à §®¬,£¨¯¥à¡®«¨ç¥áª¨© ¯ à ¡®«®¨¤ ¯®«ãç ¥âáï ¯ à ««¥«ì­ë¬ ᤢ¨£®¬ ý¯®¤¢¨¦­®©þ ¯ à ¡®«ë(2z = − y2q ,x = 0,4¢¤®«ì ý­¥¯®¤¢¨¦­®©þ ¯ à ¡®«ë (10.5).2160x2a2170x2 + a2+ yb2 = 0 | ãà ¢­¥­¨¥ ¯ àë ¬­¨¬ëå ᮯà殮­­ëå ¯¥à¥á¥ª îé¨åáﯫ®áª®á⥩.= 0 | ãà ¢­¥­¨¥ ¯ àë ¬­¨¬ëå ¯ à ««¥«ì­ëå ¯«®áª®á⥩.“à ¢­¥­¨ï ¯¯.

10 {170 (á¬. «¥ªæ¨î ü9) ­ §ë¢ îâáï ª ­®­¨ç¥áª¨¬¨ ãà ¢­¥­¨ï¬¨¯®¢¥àå­®á⥩ 2-£® ¯®à浪 .«¥¬¥­â à­ë¥ ᢮©á⢠ ¯®¢¥àå­®á⥩ 2-£® ¯®à浪 „«ï ãà ¢­¥­¨ï F (x, y, z ) = 0 ¯®¢¥àå­®á⨠2-£® ¯®à浪  ¢¢¥¤¥¬ ®¡®§­ ç¥­¨ïA1a11=  a12a13a12a22a23’®£¤  F (x, y, z ) =a11aA =  12a13a1a13a23  ,a33(xyz1)‚¢¥¤¥¬ ®¡®§­ ç¥­¨¥ϕ(x, y, z ) =(xa11 a12a13a1a12a22a23a2a13a23a33a3a1a2 ,a3a0 ,a12a22a23a2a13a23a33a3 a1xa2   y    = 0.a3za01z ) a11 a12a13ya12a22a23 = det A,δ= det A1 . a13xa23   y  .a33z áᬮâਬ  ä䨭­ãî § ¬¥­ã ª®®à¤¨­ â  xc11 y  =  c21zc31c12c22c32 0   0  xc11 c12 c13 c1xc1c13xccc   y0 y  cc23   y 0 + c2  ⇔   =  21 22 23 2   0  ,zc31 c32 c33 c3zc33z0c310 0 0 11c11 c12 c13 c1c11 c12 c13 c21 c22 c23 c2 0 = C ,  c21 c22 c23  = C1 , (a1)c31 c32 c33 c3c31 c32 c330001£¤¥ det C1 6= 0 ⇒ det C 0 6= 0 (det C1 = det C 0 ).

’®£¤ F 0 (x0 , y 0 , z 0 )= F (c11 x0 + c12 y0 + c13 z 0 + c1 , c21 x0 + c22 y0 + c23 z 0 + c2 , c31 x0 + c32 y0 + c33 z 0 + c3 )= ( x0y0z0x0y0 1 ) A0  0 .z1(a2)5¥á«®¦­® ¢¨¤¥âì, çâ®A0= (C 0 )’ AC,A01= (C1 )’ A1 C1 .¥á«®¦­® ¯®­ïâì, çâ® ¬ âà¨æ  A01 ï¥âáï ᨬ¬¥âà¨ç¥áª®©. „¥©á⢨⥫쭮, â ªª ª A1 = (A1 )’ , â®h(C1 )’ A1 C1 x, yi = hx, ((C1 )’ A1 C1 )’ yi = hx, (C1 )’ A1 C1 yi¤«ï «î¡ëå ¢¥ªâ®à®¢ x, y ∈ R3 . ( ¯®¬­¨¬, çâ® n × n-¬ âà¨æ  S ï¥âáï ᨬ¬¥âà¨ç¥áª®©, ¥á«¨ hSx, yi = hx, Syi ∀x, y ∈ Rn .) Šà®¬¥ ⮣®, â ª ª ª det C1 6= 0,â®222222A01 = C1’ A1 C1 ⇒ a0 11 + a0 22 + a0 33 + a0 12 + a0 23 + a0 13 6= 0.’ ª¨¬ ®¡à §®¬, ¨¬¥¥â ¬¥áâ® á«¥¤ãî饥’¥®à¥¬  10.2.

10 à¨ «î¡®©  ä䨭­®© § ¬¥­¥ ª®®à¤¨­ â ãà ¢­¥­¨¥ ¯®¢¥àå­®áâ¨2-£® ¯®à浪  ®áâ ­¥âáï ãà ¢­¥­¨¥¬ ¯®¢¥àå­®á⨠2-£® ¯®à浪 ;20 ¢¥«¨ç¨­ë δ , ïîâáï ®à⮣®­ «ì­ë¬¨ ¨­¢ à¨ ­â ¬¨;30 ¢¥«¨ç¨­ë sgn δ , sgn ïîâáï  ä䨭­ë¬¨ ¨­¢ à¨ ­â ¬¨.’¥®à¥¬  10.3. à¨ ­¥¯ãá⮬ ¯¥à¥á¥ç¥­¨¨ ¯®¢¥àå­®á⨠2-£® ¯®à浪  á ¯«®áª®-áâìî P ¬®£ã⠯।áâ ¢«ïâìáï á«¥¤ãî騥 á«ãç ¨{ ªà¨¢ ï 2-£® ¯®à浪  ­  ¯«®áª®á⨠P ,{ ¯àï¬ ï ­  ¯«®áª®á⨠P ,{ ¯®¢¥àå­®áâì à á¯ ¤ ¥âáï ­  ¤¢¥ ¯«®áª®áâ¨, ®¤­  ¨§ ª®â®àëå ï¥âáï ¯«®áª®áâì P , ¢å®¤ïé ï â ª¨¬ ®¡à §®¬ ¢ á®áâ ¢ à áᬠâਢ ¥¬®© ¯®¢¥àå­®á⨠.„®ª § â¥«ìá⢮.

ˆá¯®«ì§ãï § ¬¥­ã ¯¥à¥¬¥­­ëå (a1), ¯¥à¥©¤¥¬ ª â ª®© ¯àאַ㣮«ì­®© á¨á⥬¥ ª®®à¤¨­ â (x­ , y­ , z­ ), çâ® ¢ í⮩ á¨á⥬¥ ª®®à¤¨­ â ãà ¢­¥­¨¥¯«®áª®á⨠P áãâì z­ = 0. “à ¢­¥­¨¥ ¯®¢¥àå­®á⨠2-£® ¯®à浪  ¨¬¥¥â ¢ ­®¢ë媮®à¤¨­ â å á«¥¤ãî騩 ¢¨¤F ­ (x­ , y­ , z­ ) = a­11 (x­ )2 + a­22 (y­ )2 + a­33 (z­ )2+ 2a­12 x­ y­ + 2a­23 y­ z­ + 2a­13 z­ x­ + 2a­1 x­ + 2a­2 y­ + 2a­3 z­ + a­0= 0, (10.7)  ãà ¢­¥­¨¥ ¯à¥á¥ç¥­¨ï ¯®¢¥àå­®áâ¨ á ¯«®áª®áâìî § ¯¨á뢠¥âáï ª ªF ­ (x­ , y­ , 0) = a­11 (x­ )2 + a­22 (y­ )2 + 2a­12 x­ y­ + 2a­1 x­ + 2a­2 y­ + a­0…᫨ (a­11 )2 + (a­12 )2 + (a­22 )2 6= 0, â® (10.8) | ªà¨¢ ï 2-£® ¯®à浪 .= 0.(10.8)6ãáâì (a­11 )2 +(a­12 )2 +(a­22 )2 = 0, ­® (a­1 )2 +(a­2 )2 6= 0.

’®£¤  ãà ¢­¥­¨¥ ¯à¥á¥ç¥­¨ï ∩ P § ¯¨á뢠¥âáï ª ªF ­ (x­ , y­ , 0) = 2a­1 x­ + 2a­2 y­ + a­0= 0.(10.9)“à ¢­¥­¨¥ (10.9) | ãà ¢­¥­¨¥ ¯àאַ© ­  ¯«®áª®á⨠P . ’¥¯¥àì ¯ãáâì(a­11 )2 + (a­12 )2 + (a­22 )2 = (a­1 )2 + (a­2 )2 = 0.’®£¤  ¨§ (10.8) á«¥¤ã¥â, çâ® a­0 = 0, ¨ ãà ¢­¥­¨¥ (10.7) ¨¬¥¥â ¢¨¤¢¡F ­ (x­ , y­ , z­ ) = z­ a­33 z­ + 2a­23 y­ + 2a­13 x­ + 2a­3 = 0,â. ¥. ¯®¢¥àå­®áâì à á¯ ¤ ¥âáï ­  ¯ àã ¯«®áª®á⥩, ®¤­  ¨§ ª®â®àëå P (z­ = 0).¥¥ª®â®àë¥ á¢®©á⢠ ᨬ¬¥âà¨ç¥áª¨å ¬ âà¨æ‘¢®©á⢮ 10.1.

ãáâì A | n × n-ᨬ¬¥âà¨ç¥áª ï ¬ âà¨æ , CC −1 AC| n × n-ᨬ¬¥âà¨ç¥áª ï ¬ âà¨æ .„®ª § â¥«ìá⢮.C −1 AC∈ O(n).’®£¤ = C ’ AC ⇒ (C ’ AC )’ = C ’ AC = C −1 AC.¥ˆ§¢¥áâ­®, çâ® ¢á¥ ᮡá⢥­­ë¥ ç¨á«  «î¡®© ᨬ¬¥âà¨ç¥áª®© ¬ âà¨æë ¢¥é¥á⢥­­ë. ¨¦¥ ¬ë ¤®ª ¦¥¬ íâ®â 䠪⠤«ï á«ãç ï 3-ᨬ¬¥âà¨ç¥áª¨å ¬ âà¨æ.‘¢®©á⢮ 10.2. ãáâì A | 3 × 3-ᨬ¬¥âà¨ç¥áª ï ¬ âà¨æ . ’®£¤  ¢á¥ ᮡá⢥­­ë¥ ç¨á«  ¬ âà¨æë A ¢¥é¥á⢥­­ë.„®ª § â¥«ìá⢮.

• à ªâ¥à¨áâ¨ç¥áª¨© ¯®«¨­®¬ det(A − λE ) ¨¬¥¥â á⥯¥­ì 3, ¯®í⮬ã, ¢ ᨫ㠮᭮¢­®© â¥®à¥¬ë «¨­¥©­®©  «£¥¡àë, ã ãà ¢­¥­¨ï det(A − λE ) ¥áâì å®âï¡ë ®¤­® ¢¥é¥á⢥­­®¥ ç¨á«® λ0 . ãáâì C | 3 × 3-¬ âà¨æ  â ª ï, çâ® ¥¥ ¯¥à¢ë©á⮫¡¥æ v0 | ᮡá⢥­­ë© ¢¥ªâ®à ¬ âà¨æë A ¥¤¨­¨ç­®© ¤«¨­ë, ᮮ⢥âáâ¢ãî騩ᮡá⢥­­®¬ã ç¨á«ã λ0 ,   ®áâ «ì­ë¥ ¢¥ªâ®à-á⮫¡æë v1 , v2 ¬ âà¨æë A â ª®¢ë, çâ®âனª  v0 , v1 , v2 ®¡à §ã¥â ®àâ®­®à¬¨à®¢ ­­ë© ¡ §¨á ¯à®áâà ­á⢠ R3 . ’®£¤  ¯®á¢®©áâ¢ã 10.1 ¬ë ¨¬¥¥¬λ0 0 0C −1 AC =  0 a b 0 b cµ¶a b¤«ï ­¥ª®â®àëå ç¨á¥« a, b, c. ‘®¡á⢥­­ë¥ ç¨á«  λ1 , λ2 ¬ âà¨æë b c ¢¥é¥á⢥­­ë (á¬. «¥¬¬ã 4.2 ¨§ «¥ªæ¨¨ ü4). ãáâì u1 = (a1 , b1 ), u2 = (a2 , b2 ) | ᮡá⢥­­ë¥7µa bb c¶¢¥ªâ®àë ¬ âà¨æë, ᮮ⢥âáâ¢ãî騥 ᮡá⢥­­ë¬ ç¨á« ¬ λ1 , λ2 , ®¡à §ãî騥 ®àâ®­®à¬¨à®¢ ­­ë© ¡ §¨á ¯«®áª®áâ¨. ’®£¤ 1 0D = 0 a10 b1¨ ¬ë ¨¬¥¥¬0a2  ∈ O(3),b2D−1 C −1 ACDλ0000=  0 λ10 0 λ2 .’ ª¨¬ ®¡à §®¬, λ0 , λ1 , λ2 | ᮡá⢥­­ë¥ ᮡá⢥­­ë¥ ç¨á«  ¬ âà¨æë A, ᮡá⢥­­ë¥ ¢¥ªâ®àë ¨¬ ᮮ⢥âáâ¢ãî騥, | ¢¥ªâ®à-á⮫¡æë ¬ âà¨æë CD.¥‘¢®©á⢮ 10.3.

ãáâì A | ᨬ¬¥âà¨ç¥áª ï (n × n)-¬ âà¨æ , λ1 , . . . , λk | ¢á¥¥¥ à §«¨ç­ë¥ ᮡá⢥­­ë¥ ç¨á« , Vi | ¢¥ªâ®à­®¥ ¯®¤¯à®áâà ­á⢮, ­ âï­ã⮥­  ᮡá⢥­­ë¥ ¢¥ªâ®àë, ®â¢¥ç î騥 ᮡá⢥­­®¬ã ç¨á«ã λi . ’®£¤  Rn = V1 ⊕· · · ⊕ Vk , £¤¥ Rn | áâ ­¤ àâ­®¥ ¥¢ª«¨¤®¢® ¯à®áâà ­á⢮.„®ª § â¥«ìá⢮. Ž¡®§­ ç¨¬ V = V1 ⊕· · ·⊕Vk (áã¬¬ë §¤¥áì ¯àï¬ë¥ ¢ ᨫ㠫¥¬¬ë 4.1«¥ªæ¨¨ ü4). à¥¤¯®«®¦¨¬, çâ® V ⊥ 6= {0}. ãáâì ui1 , .