L-4-Spring2018 (Грешнов - Лекции (1-15))

‹…Š–ˆŸ ü4„ â : 27.02.2018’¥®à¥¬  4.1. „«ï «î¡®£® ç¨á«  ² > 0, ¯àאַ© δ ¨ â®çª¨ M áãé¥áâ¢ã¥â ¥¤¨­-á⢥­­ ï £¨¯¥à¡®«  (² > 1), ¯ à ¡®«  (² = 1), í««¨¯á (² < 1) á íªá業âà¨á¨â¥â®¬², ¯à ¢ë¬ 䮪ãᮬ M ¨ ᮮ⢥âáâ¢ãî饩 ¥¬ã ¤¨à¥ªâà¨á®© ¤¨à¥ªâà¨á®© δ .„®ª § â¥«ìá⢮. ãáâì ­  ¥¢ª«¨¤®¢®© ¯«®áª®á⨠¤ ­ë ¯àﬠï δ ¨ â®çª  M â ª¨¥,çâ® M ∈/ δ . ’®£¤  ¬ë ¬®¦¥¬ ®¯à¥¤¥«¨âì à ááâ®ï­¨¥ d(M, δ ) = f . à¥¤¯®«®¦¨¬,­¥ 㬥­ìè ï ®¡é­®áâ¨, çâ® â®çª  M ­ å®¤¨âáï «¥¢¥¥ (¢ ®¡ëç­®¬ á¬ëá«¥) ¯àאַ©δ.à®¢¥¤¥¬ ¤®ª § â¥«ìá⢮ ¤«ï á«ãç ï ² ∈ (0, 1), â. ¥. ®¯à¥¤¥«¨¬ í««¨¯á á íªá業âà¨á¨â¥â®¬ ², ¯à ¢ë¬ 䮪ãᮬ F¯ = M ¨ ¯à ¢®© ¤¨à¥ªâà¨á®© δ¯ = δ .

„«ï í⮣®¢ë¡¥à¥¬ ­  ¯«®áª®á⨠­¥ª®â®àãî ¯àאַ㣮«ì­ãî á¨á⥬㠪®®à¤¨­ â â ª, çâ® â®çª M ¯à¨­ ¤«¥¦¨â ®á¨  ¡æ¨áá,   ®áì ®à¤¨­ â ¯ à ««¥«ì­  ¯àאַ© δ . ˆ§ ®¯à¥¤¥«¥­¨ï ¤¨à¥ªâà¨á í««¨¯á  ¢ë⥪ ¥â, çâ® ¢ ¢ë¡à ­­®© á¨á⥬¥ ª®®à¤¨­ â ãà ¢­¥­¨¥¤¨à¥ªâà¨áë δ¯ = δ ¨¬¥¥â ¢¨¤ x = a² ,   à ááâ®ï­¨¥ ®â â®çª¨ F¯ = M ¤® ¤¨à¥ªâà¨áëδ¯ = δ à ¢­® a² − a²; §¤¥áì ¡®«ìè ï ¯®«ã®áì a ¯®ª  çâ® ­¥ ®¯à¥¤¥«¥­ . Œë ¨¬¥¥¬d(M, δ ) = d(F¯ , δ¯ ) =a− a² = f ⇔ f²=a1 − ²2²⇔a=f².1 − ²2’ ª¨¬ ®¡à §®¬, ¬ë ®¯à¥¤¥«¨«¨ §­ ç¥­¨¥ ¡®«ì让 ¯®«ã®á¨ a. ˆáå®¤ï ¨§ ⮦¤¥á⢠√√a² = c = a2 − b2 , ¬ë ®¯à¥¤¥«ï¥¬ §­ ç¥­¨¥ ¬ «®© ¯®«ã®á¨ í««¨¯á  b = a 1 − ²2 .√’®£¤  ¬ë ­ å®¤¨¬ §­ ç¥­¨¥ «¨­¥©­®£® íªá業à¨á¨â¥â  c = a2 − b2 , ®âªã¤  ¯®«ãç ¥¬, çâ® à ááâ®ï­¨¥ 業âà  í««¨¯á  O (­ ç «® ­ è¥© ¯àאַ㣮«ì­®© á¨á⥬몮®à¤¨­ â) ¤® ¯àאַ© δ à ¢­® c + f . ’ ª¨¬ ®¡à §®¬, ¬ë ®¯à¥¤¥«¨«¨ ¬¥áâ® à á¯®«®¦¥­¨ï 業âà  í««¨¯á  ­  ¯àאַ©, ¯¥à¯¥­¤¨ªã«ïà­®© ¯àאַ© δ , ­ è«¨ ¯®«ã®á¨í««¨¯á ; ¯à¨ í⮬ δ = δ¯ , M = F¯ ,   ç¨á«® ² | íªá業âà¨á¨â¥â í««¨¯á .®áâ஥­¨¥ £¨¯¥à¡®«ë (² > 1) ¨«¨ ¯ à ¡®«ë (² = 1) ¯à®¨á室¨â  ­ «®£¨ç­®.

¥C¢®©á⢮ 4.1. Œ­®¦¥á⢮ â®ç¥ª ¯«®áª®áâ¨, ¨§ ª®â®àëå í««¨¯á ¢¨¤¥­ ¯®¤ ¯àï-¬ë¬ 㣫®¬, ï¥âáï ®ªà㦭®áâìî á 業â஬ ¢ ­ ç «¥ ª®®à¤¨­ â à ¤¨ãá √a2 + b2 .„®ª § â¥«ìá⢮.  áᬮâਬ ¯àï¬ë¥ lX , lY , ª á â¥«ì­ë¥ ª í««¨¯áã ¢ â®çª å í««¨¯á  X, Y , lX ∩lY = P . ãáâì Fe« , Fe¯ | â®çª¨, ᨬ¬¥âà¨ç­ë¥ 䮪ãá ¬ í««¨¯á  F« , F¯ ®â­®á¨â¥«ì­® ¯àï¬ëå lX , lY ᮮ⢥âá⢥­­®. ˆ§ £¥®¬¥âà¨ç¥áª®£®®¯à¥¤¥«¥­¨ï í««¨¯á  á«¥¤ã¥â, çâ®d(F« , X ) + d(X, F¯ ) = d(F« , Y ) + d(Y, F¯ ) = 2a⇒ d(Fe« , X ) + d(X, F¯ ) = d(F« , Y ) + d(Y, Fe¯ ) = 2a.12ˆ§ ®¯â¨ç¥áª®£® ᢮©á⢠ í««¨¯á  ¢ë⥪ ¥â, çâ® â®çª¨ Fe« , X, F¯ «¥¦ â ­  ®¤­®© ¯àאַ©,   â ª¦¥ â®çª¨ F« , Y, Fe¯ «¥¦ â ­  ®¤­®© ¯àאַ©, á«¥¤®¢ â¥«ì­®, d(Fe« , F¯ ) =d(F« , Fe¯ ) = 2a.

’®£¤ , ãç¨â뢠ï, çâ® d(Fe« , P ) = d(F« , P ), d(Fe¯ , P ) = d(F¯ , P ), ¬ë¯®«ãç ¥¬, çâ® 4Fe« P F¯ = 4F« P Fe¯ (¯à¨§­ ª à ¢¥­á⢠ âà¥ã£®«ì­¨ª®¢ ¯® â६ áâ®à®­ ¬). ’®£¤  ᮮ⢥âáâ¢ãî騥 ã£«ë ¢ âà¥ã£®«ì­¨ª å 4Fe« P F¯ , 4F« P Fe¯ à ¢­ë,®âªã¤ 2∠XP F« + ∠F« P F¯ = 2∠Y P F¯ + ∠F¯ P F« ⇒ ∠XP F« = ∠F¯ P Y.Œë å®â¨¬ ­ ©â¨ £¥®¬¥âà¨ç¥áª®¥ ¬¥áâ® â®ç¥ª P â ª¨å, çâ® ∠XP Y = π2 . ˆ§¨§®£®­ «ì­®£® ᢮©á⢠ í««¨¯á  ¢ë⥪ ¥â, çâ® ∠XP Y = ∠Fe« P F¯ . ˆ§ £¥®¬¥âà¨ç¥áª®£® ®¯à¥¤¥«¥­¨ï í««¨¯á  ¬ë ¨¬¥¥¬ d(Fe« , F¯ ) = d(F« , X ) + d(X, F¯ ) = 2a.

’ ª ª ª∠Fe« P F¯ = π2 , â® ¯® ⥮६¥ ¨ä £®à  ¬ë ¯®«ãç ¥¬(2a)2 = d(Fe« , F¯ )2 = d(Fe« , P )2 + d2 (P, F¯ ) ⇒ (2a)2 = d(F« , P )2 + d2 (P, F¯ ). (4.1)“á«®¢¨¥ (4.1) ®¯à¥¤¥«ï¥â ᮡ®© ãà ¢­¥­¨¥ ®ªà㦭®á⨠á 業â஬ ¢ ­ ç «¥ ª®®à¤¨­ â. „¥©á⢨⥫쭮, ¯ãáâì P = (x, y), ⮣¤  (4.1) íª¢¨¢ «¥­â­® à ¢¥­áâ¢ã(x + c)2 + y2 + (x − c)2 + y2 = 4a2 ⇔ 2x2 + 2y2 = 4a2 − 2c2 = 2a2 + 2b2 .¥‡ ¤ ç  4.1.  ©â¨ £¥®¬¥âà¨ç¥áª®¥ ¬¥áâ® â®ç¥ª ¯«®áª®áâ¨, ¨§ ª®â®àëå ¯ à ¡®« ¨ £¨¯¥à¡®«  ¢¨¤­ë ¯®¤ ¯àï¬ë¬ 㣫®¬.««¨¯á, £¨¯¥à¡®« , ¯ à ¡®«  ¢ ¯®«ïà­®© á¨á⥬¥ ª®®à¤¨­ â áᬮâਬ ¯®«ïà­ãî á¨á⥬㠪®®à¤¨­ â á ¯®«îᮬ ¢ â®çª¥ O, ᮢ¯ ¤ î饩 áâ®çª®©{ F« ¢ á«ãç ¥ í««¨¯á  («¥¢ë© 䮪ãá í««¨¯á ),{ F¯ ¢ á«ãç ¥ £¨¯¥à¡®«ë (¯à ¢ë© 䮪ãá í««¨¯á ),{ F ¢ á«ãç ¥ ¯ à ¡®«ë (䮪ãá ¯ à ¡®«ë),¨ ¯®«ïà­®© ®áìî, ¯à¨­ ¤«¥¦ é¥© ¢ á«ãç ¥ í««¨¯á  ¨ £¨¯¥à¡®«ë ¯àאַ©, ¯à®å®¤ï饩 ç¥à¥§ ¨å 䮪ãáë,   ¢ á«ãç ¥ ¯ à ¡®«ë | ®á¨ ᨬ¬¥âਨ ¯ à ¡®«ë.

‘¨¬¢®«®¬δ ¬ë ¡ã¤¥¬ §¤¥áì ®¡®§­ ç âì ¢ á«ãç ¥ í««¨¯á  ¥£® «¥¢ãî ¤¨à¥ªâà¨áã, ¢ á«ãç ¥ £¨¯¥à¡®«ë | ¥£® ¯à ¢ãî ¤¨à¥ªâà¨áã, ¤¨à¥ªâà¨áã | ¢ á«ãç ¥ £¨¯¥à¡®«ë. ãáâì M |¯à®¨§¢®«ì­ ï â®çª  í««¨¯á , ¯ à ¡®«ë ¨«¨ ¯à ¢®© ¢¥â¢¨ £¨¯¥à¡®«ë. ˆá¯®«ì§ãï«¥¬¬ã 3.3 ¨§ «¥ªæ¨¨ ü3, ¬ë ¬®¦¥¬ § ¯¨á âìd(M, O)d(M, δ )= ² ⇔ d(M, δ) =d(M, O).²3Šà®¬¥ ⮣®, ¬ë ¨¬¥¥¬ d(O, δ ) = p² , £¤¥ p | 䮪 «ì­ë© ¯ à ¬¥âà.

Ž¡®§­ ç¨¬r = d(O, M ) (¤«¨­  à ¤¨ãá-¢¥ªâ®à  ⥪ã饩 â®çª¨ M ). ’®£¤ r²p²= d(M, δ) = + r cos ϕ ⇔ r = p + r² cos ϕ ⇔ r =p.1 − ² cos ϕ(4.2)®á«¥¤­¥¥ ⮦¤¥á⢮ ¢ (4.2) ï¥âáï ãà ¢­¥­¨¥¬ ¢ ¯®«ïà­ëå ª®®à¤¨­ â å{ í««¨¯á , ² < 1, ϕ ∈ [0, 2π),{ ¯ à ¡®«ë, ² = 1, ϕ ∈ (0, 2π), ¯à¨ ϕ = 0, 2𠤫¨­  à ¤¨ãá-¢¥ªâ®à  â®çª¨ Mà ¢­  ∞.Ž¯à¥¤¥«¨¬ ®¡« áâì §­ ç¥­¨© ¯ à ¬¥âà  ϕ ¤«ï ¯à ¢®© ¢¥â¢¨ £¨¯¥à¡®«ë (² > 1). ¯®¬­¨¬, çâ® ¯àï¬ë¥ y = ± ab x ïîâáï  á¨¬¯â®â ¬¨ £¨¯¥à¡®«ë. ®í⮬ãsup k = ab (inf k = − ab ), £¤¥ k ॣ« ¬¥­â¨àã¥âáï ⥬ ãá«®¢¨¥¬, çâ® «ãç, ¯à¨­ ¤«¥¦ é¨© ¯àאַ© y = c + kx, ¨á室ï騩 ¨§ â®çª¨ O, ¯à¨­ ¤«¥¦¨â ¯à ¢®¬ã ¢¥àå­¥¬ã(­¨¦­¥¬ã) 㣫㠪®®à¤¨­ â­®© ¥¢ª«¨¤®¢®© ¯«®áª®á⨠¨ ­¥ ¯¥à¥á¥ª ¥â ¯à ¢ãî ¢¥â¢ì£¨¯¥à¡®«ë. ’. ¥. ¢ á«ãç ¥ £¨¯¥à¡®«ëϕ ∈ (ϕ0 , 2π − ϕ0 ),ϕ0= arctgb.a(4.3)Œë ¨¬¥¥¬b2a2= tg2 ϕ0 =1 − cos2 ϕ01= 22cos ϕ0cos ϕ0−1⇒11= 22cos ϕ0 ²¡⇒ ϕ0= arccos1²,¢â.

¥. (4.3) ¬®¦­® § ¯¨á âì ª ª ϕ ∈ arccos 1² , 2π − arccos 1² ; ®â¬¥â¨¬, çâ® ¯à¨ ϕ =arccos 1² , 2π − arccos 1² ¤«¨­  à ¤¨ãá-¢¥ªâ®à  â®çª¨ M à ¢­  ∞.“¯à ¦­¥­¨¥ 4.1. à¨ ª ª¨å §­ ç¥­¨ïå ¯ à ¬¥â஢ a, b, c ãà ¢­¥­¨¥ ¢ ¯®«ïà­ë媮®à¤¨­ â å r =c1+a cos ϕ+b sin ϕï¥âáï í««¨¯á®¬, £¨¯¥à¡®«®©, ¯ à ¡®«®©?Š« áá¨ä¨ª æ¨ï ªà¨¢ëå 2-£® ¯®à浪  áᬮâਬ ®¡é¥¥ ãà ¢­¥­¨¥ ªà¨¢ëå 2-£® ¯®à浪 F (x, y ) = a11 x2 + 2a12 xy + a22 y 2 + 2a1 x + 2a2 y + a0= 0,(4.4)£¤¥ å®âï ¡ë ®¤­® ¨§ ç¨á¥« a11 , a12 , a22 ­¥ à ¢­® ­ã«î.

‚¢¥¤¥¬ ­  ¯«®áª®á⨠­®¢ë¥¯àאַ㣮«ì­ë¥ ª®®à¤¨­ âë (x­ , y­ ) ¯ã⥬ § ¬¥­ëµcos ϕsin ϕ− sin ϕcos ϕ¶µx­y­¶µ ¶= xy .4’®£¤  ¬ë ¬®¦¥¬ § ¯¨á âìa11 x2 + 2a12 xy + a22 y 2= a11 (x­ cos ϕ − y­ sin ϕ)2 + 2a12 (x­ cos ϕ − y­ sin ϕ)(x­ sin ϕ + y­ cos ϕ)¢−2a11 sin ϕ cos ϕ+2a22 sin ϕ cos ϕ+2a12 (cos2 ϕ−sin2 ϕ)¡¢+ h1 x2­ + h2 y­2 = x­ y­ (a22 − a11 ) sin 2ϕ + 2a12 cos 2ϕ + h1 x2­ + h2 y­2 . (4.5)+a22 (x­ sin ϕ+y­ cos ϕ)2 = x­ y­¡ãáâì a12 6= 0, ⮣¤  ¬ë ¢á¥£¤  ¬®¦¥¬ ­ ©â¨ 㣮« ϕ â ª®©, çâ® ª®íää¨æ¨¥­â ¯à¨x­ y­ ¢ (4.5) à ¢­ï«áï ¡ë ­ã«î; ¤¥©á⢨⥫쭮, ¢ í⮬ á«ãç ¥ 㣮« ϕ ®ç¥¢¨¤­®®¯à¥¤¥«ï¥âáï ¨§ á«¥¤ãî饩 ä®à¬ã«ë â ­£¥­á  㤢®¥­­®£® 㣫 tg 2ϕ =2a12a11 − a22,ϕ ∈ (0, π/2].(4.6)®í⮬ã, ¯¥à¥å®¤ï ¯à¨ ­¥®¡å®¤¨¬®á⨠ª ¤à㣨¬ ª®®à¤¨­ â ¬ ¯ã⥬ ¯®¢®à®â  ª®®à¤¨­ â­ëå ®á¥© ­  ᮮ⢥âáâ¢ãî騩 㣮« ϕ ¨§ (4.6), ¬ë ¢á¥£¤  ¬®¦¥¬ à áᬠâਢ âì ãà ¢­¥­¨¥a11 x2 + a22 y 2 + 2a1 x + 2a2 y + a0 = 0(4.7)¢¬¥áâ® ãà ¢­¥­¨ï (4.4). áᬮâਬ ãà ¢­¥­¨¥ (4.7).‘«ãç © I (業âà «ì­ë© á«ãç ©).¤¨­ âë ¯® ä®à¬ã« ¬x~=x+a11 a22 6=a1,a110.

‚ í⮬ á«ãç ¥ ýᤢ¨­¥¬þ ª®®à-y~ = x +a2,a22¢ १ã«ìâ â¥ ãà ¢­¥­¨¥ (4.7) ¯à¨¬¥â ¢¨¤³³a1 ´2a ´22a2 2a2a11 x ++ a22 y + 2 + a0 − 1 − 2a11a22a11a22= a11 x~2 + a22 y~2 + c = 0,c = const .(4.8)‚ § ¢¨á¨¬®á⨠®â ª®íää¨æ¨¥­â®¢ a11 , a22 , c ãà ¢­¥­¨¥ (4.8) ¯à¥¤áâ ¢«ï¥â ­  ᮡ®î½a11 a22 > 0,0| í««¨¯á,1 c 6= 0,a11 c < 020 c 6= 0, a11 a22 < 0, | £¨¯¥à¡®« ,30c 6= 0,sgn a11 = sgn a22 = sgn c | ¬­¨¬ë© í««¨¯á,40c = 0,a11 a22 < 050 c = 0 ,¤¨­ â.a11 a22 >| ¤¢¥ ¯¥à¥á¥ª î騥áï ¯àï¬ë¥,0 | ¤¢¥ ¬­¨¬ë¥ ¯àï¬ë¥, ¯¥à¥á¥ª î騥áï ¢ ­ ç «¥ ª®®à-5‘«ãç © II (¯ à ¡®«¨ç¥áª¨© á«ãç ©).= 0. à¥¤¯®«®¦¨¬, ­¥ 㬥­ìè ï®¡é­®áâ¨, çâ® a11 = 0. ’®£¤  ãà ¢­¥­¨¥ (4.7) ¯à¨¬¥â ¢¨¤a11 a22a22 y 2 + 2a1 x + 2a2 y + a0= 0.(4.9)ý‘¤¢¨­¥¬þ ª®®à¤¨­ âë ¯® ä®à¬ã« ¬x~ = x,y~ = y +a2.a22‚ १ã«ìâ â¥ ãà ¢­¥­¨¥ (4.9) ¯à¨¬¥â ¢¨¤³a2 ´2a2a22 y ++ 2a 1 x + a 0 − 2a22a22= a22 y~2 + 2a1 x~ + c = 0,c = const .(4.10)ˆáá«¥¤ã¥¬ ãà ¢­¥­¨¥ (4.10) ¢ § ¢¨á¨¬®á⨠®â ª®íää¨æ¨¥­â®¢ a22 , a1 , c.a1 6= 0,⮣¤  ¯¥à¥å®¤¨¬ ¢ (4.10) ª ª®®à¤¨­ â ¬ x^ = x~ + 2ac 1 , y^ = y~, ¨ ¯®«ãç ¥¬ãà ¢­¥­¨¥ ¯ à ¡®«ë60 y^2 = − 2aa221 x^,= 0, ⮣¤  ãà ¢­¥­¨¥ (4.10) ¨¬¥¥â ¢¨¤ a22 y~2 + c = 0, ª®â®à®¥ ¢ á¢®î ®ç¥à¥¤ìà §¡¨¢ ¥âáï ­  á«ãç ¨70 a22 c < 0 | ¤¢¥ ¯ à ««¥«ì­ë¥ ¯àï¬ë¥,80 a22 c > 0 | ¤¢¥ ¬­¨¬ë¥ ¯àï¬ë¥,90 a22 c = 0 ⇔ y~2 = 0 | ¤¢¥ ᮢ¯ ¤ î騥 ¯àï¬ë¥.a1“à ¢­¥­¨ï 10 {90 ¯à¥¤áâ ¢«ïîâ ᮡ®© ª ­®­¨ç¥áª¨¥ ä®à¬ë ªà¨¢ëå 2-£® ¯®à浪 .à¨¢¥¤¥­¨¥ ªà¨¢®© 2-£® ¯®à浪  ª ª ­®­¨ç¥áª®¬ã ¢¨¤ã.Ž¯à¥¤¥«¥­¨¥ 4.1.

(n × n)-¬ âà¨æ  S ­ §ë¢ ¥âáï ᨬ¬¥âà¨ç¥áª®©, ¥á«¨ S = S ’ .à¨¬¥à 4.1. „«ï «î¡®© (n × n)-¬ âà¨æë A ¬ âà¨æ  A’ A ᨬ¬¥âà¨ç¥áª ï.‹¥¬¬  4.1. ãáâì S | ᨬ¬¥âà¨ç¥áª ï ¬ âà¨æ . ’®£¤ :10 ᮡá⢥­­ë¥ ¢¥ªâ®àë, ®â¢¥ç î騥 à §­ë¬ ᮡá⢥­­ë¬ ç¨á« ¬ ᨬ¬¥âà¨ç¥áª®£® ¯à¥®¡à §®¢ ­¨ï f , ®à⮣®­ «ì­ë;20 ¢á¥ ᮡá⢥­­ë¥ ¢¥ªâ®àë, ®â¢¥ç î騥 ®¤­®¬ã ¨ ⮬㠦¥ ᮡá⢥­­®¬ãç¨á«ã λ ¬ âà¨æë S , ¨ ­ã«¥¢®© ¢¥ªâ®à 0 ®¡à §ãîâ ¢¥ªâ®à­®¥ ¯®¤¯à®áâà ­á⢮.„®ª § â¥«ìá⢮. 10 ãáâì Su1 = λ1 u1 , Su2 = λ2 u2 , λ1 6= λ2 .

’®£¤ λ1 hu1 , u2 i = hλ1 u1 , u2 i= hf (u1 ), u2 i = hu1 , f (u2 )i = hu1 , λ2 u2 i = λ2 hu1 , u2 i ⇒ hu1 , u2 i = 0.620 Œë ¨¬¥¥¬ S (0) = λ0 = 0; ¥á«¨ Sv = λv, â® ®ç¥¢¨¤­® S (kv) = λkv ∀k ∈ R; ¥á«¨Sv = λv , Su = λu, â® ®ç¥¢¨¤­® S (u + v ) = Su + Sv = λu + λv = λ(u + vµ).¥¶a11 a12‹¥¬¬  4.2. ‚ᥠᮡá⢥­­ë¥ ç¨á«  ᨬ¬¥âà¨ç¥áª®© ¬ âà¨æë A1 = a a1222¢¥é¥á⢥­­ë.„®ª § â¥«ìá⢮ Œë ¨¬¥¥¬A1 u = λu ⇔ (A1 − λE )u = 0 ⇔ det(A1 − λE ) = 0⇔ λ2 − (a11 + a22 )λ + a11 a22 − a212= 0 ⇔ λ2 − Sλ + δ = 0, (4.11)£¤¥ S = a11 + a22 | á«¥¤ ¬ âà¨æë A1 , δ = det A1 .

Š®à­¨ ª¢ ¤à â­®£® ãà ¢­¥­¨ï(4.11) ¨¬¥îâ ¢¨¤λ1 , 2=S±√S 2 − 4δ2=S±p(a11 − a22 )2 + 4a212,2®âªã¤  ¨ á«¥¤ã¥â «¥¬¬  4.2.‘«¥¤á⢨¥ 4.1. à¨ ãá«®¢¨¨ a211 + a212 + a222 6= 0 ¬ë ¨¬¥¥¬ λ21 + λ22 6= 0.„®ª § â¥«ìá⢮. ‘«¥¤á⢨¥ 4.1 ¢ë⥪ ¥â ¨§ (4.12).‘«¥¤á⢨¥ 4.2. λ1 + λ2 = S , λ1 λ2 = δ.„®ª § â¥«ìá⢮. ‘«¥¤á⢨¥ 4.1 ¢ë⥪ ¥â ¨§ (4.12).‚¢¥¤¥¬ ­®¢ë¥ ¯¥à¥¬¥­­ë¥µcos αsin α− sin αcos ᶵx­y­¶(4.12)¥¥¥µ ¶= xy ,£¤¥ u = (cos α, sin α), v = (− sin α, cos α) | ®àâ®­®à¬¨à®¢ ­­ë© ¡ §¨á ¨§ ᮡá⢥­­ëå ¢¥ªâ®à®¢ ¬ âà¨æë A1 (¤à㣨¬¨ á«®¢ ¬¨, ¯®¢¥à­¥¬ ª®®à¤¨­ â­ë¥ ®á¨ ­  㣮«α ¯à®â¨¢ ç á®¢®© áâ५ª¨).

’®£¤ ¶µ ¶a11 a12xy)a12 a22y¶µ¶µ¶µ ¶sin αa11 a12cos α − sin αx­cos αa12 a22sin α cos αy­µ¶µ ¶x­( x­ y­ ) λ01 λ0= λ1 (x­ )2 + λ2 (y­ )2 .y­2a11 x2 + 2a12 xy + a22 y 2 = ( xµα= ( x­ y­ ) −cossin αµ(4.13)ˆá¯®«ì§ãï (4.13), ¬ë ¯®«ãç ¥¬, çâ® ãà ¢­¥­¨¥ (4.4) ¢ á¨á⥬¥ ª®®à¤¨­ â (x­ , y­ )¨¬¥¥â ¢¨¤λ1 (x­ )2 + λ2 (y­ )2 + 2x­ (a1 cos α + a2 sin α) + 2y­ (a2 cos α − a1 sin α) + a0= λ1 (x­ )2 + λ2 (y­ )2 + 2x­ a01 + 2y­ a02 + a0 = 0.(4.14).