矩阵

㆛ 㽕

㆛ ЎϸϾ䚼 DŽ

㄀ϔ䚼 ⬅㄀6ˈ7ˈ8ˈ9ゴ DŽ㄀6ˈ7ˈ8ゴЏ㽕ҟ㒡 ⶽ䰉ⶹ䆚DŽⶽ䰉⧚䆎Ϣㅫ⊩ 㾷 䅵ㅫ 㾚㾝 䯂乬ⱘ ˈ ҹ䇈Ңџ䆹乚 ⱘⷨおҎ 䛑䗮 䖭䮼 DŽ ㆛ Ў䙷ѯ Ңџ䅵ㅫ 㾚㾝ⷨおⱘҎ կ 㽕ⱘⶽ䰉ⶹ䆚DŽ 䞣ҷ Ϣ ⦄ҷ⧚䆎⠽⧚ ⱘ䞡㽕 ˈ䖥 ҎӀ (Џ㽕 䞣ҷ ) 䅵ㅫ 㾚㾝 ⫼ 㾷 䯂乬ˈ ㄀9ゴ Ӏҟ㒡њ 䞣ҷ ⱘⶹ䆚DŽ

㄀Ѡ䚼 ⬅㄀10ˈ11ˈ12ゴ ˈЏ㽕ҟ㒡ⶽ䰉Ϣ 䞣 䅵ㅫ 㾚㾝Ёⱘ Ͼ ⫼DŽ 䰙Ϟˈ ㄀ϔ㆛ 㒣ⳟ њⶽ䰉⧚䆎 ⧚ϝ㓈䅵ㅫ 㾚㾝䯂乬ⱘ԰⫼ˈ ㆛ Ӯ䖯ϔℹⳟ ⶽ䰉Ϣ 䞣ⱘ䞡㽕԰⫼DŽ

ゴЏ㽕 ϟ˖

㄀ ゴ ⶽ䰉ⱘℷѸ 䜝 㾦 ⧚䆎ˈЏ㽕 ⃻⇣ 䜝 ぎ䯈ϢℷѸ 䜝 ⶽ䰉ˈℷ㾘ⶽ䰉ℷѸ 䜝 㾦 ⧚䆎ˈℷ㾘ⶽ䰉ⱘℷѸ䈅 㾷 ⓨㅫDŽ

㄀ ゴ ⶽ䰉 㾷⧚䆎ˈЏ㽕 ⶽ䰉ⱘℷѸϝ㾦 㾷ˈ ⿄ⶽ䰉Ϣ ⿄ⶽ䰉ⱘ⡍ 㾷ˈℷ ⶽ䰉ⱘШ䞠 㾷ˈҹ ⶽ䰉ⱘ69' 㾷Ϣ 㾷DŽ Ӏ 䅵ㅫ 㾚㾝Ё䛑 䞡㽕ⱘ ⫼DŽ

㄀ ゴ ⶽ䰉 ⱘ ˈЏ㽕 䞣Ϣⶽ䰉㣗 ˈ ѠЬ䯂乬ⱘ69' 㾷 ⊩ˈⶽ䰉㑻 Ϣⶽ䰉 ˈҹ ⶽ䰉ⱘ Ϣ⿃ DŽ Ӏ䛑 䅵ㅫ 㾚㾝Ёⱘ DŽ ㄀9ゴ 䞣ҷ ˈЏ㽕 䞣⿃ˈ 䞣Ϣ ҷ ⱘϔѯ ὖ DŽ 䞣ὖ ⦄ 䅵ㅫ 㾚㾝ⱘ ⷨお ⤂Ёˈ ゴⱘЏ㽕Ⳃⱘ Ў Ңџ䅵ㅫ 㾚㾝ⷨおⱘҎ կ 䞣 ⶹ䆚DŽ

㄀10ゴ ⶽ䰉 ⊩ ⹂ Ёⱘ ⫼DŽЏ㽕ҟ㒡∖㾷 ⶽ䰉ⱘSVD ⊩ ⬅ ⶽ䰉㦋 ǃ ⱘQR 㾷 ⊩ˈ 䖬ҟ㒡њ∖㾷㒣 N⚍䗣㾚䯂乬(PNP䯂乬)ⱘⶽ䰉 ⊩DŽ

㄀11ゴ ⶽ䰉SVD 㾷 䅵ㅫ 㾚㾝ЁⱘϸϾ ⫼DŽ䖤 Ԅ䅵ǃ 䞡 ⱘSVD 㾷 ⊩DŽ

㄀12ゴ 䞣 㾚⚍ ԩЁⱘ ⫼DŽЏ㽕ҟ㒡 䗄 䞡㒓 ㋏ⱘ 䞣ˈ䖭ѯ 䞣 ϡ 䞣ˈ Ӏ 䖥 䍋 ⱘ 㾚⚍ ԩ DŽ

⽣ ˖䅵ㅫ 㾚㾝Ёⱘ ⊩ , ㄀ ゴ˖ⶽ䰉ⱘ䜝 ℷѸ 㾦 118

㄀ ゴ

㈹䎔⭥䇰 䎞ㅜ ⰵㅨ⿐

6.1 ⿃ぎ䯈ϢℷѸⶽ䰉

6.1.1 ⿃ぎ䯈

Н6.1.1䆒V R Ϟⱘ㒓 ぎ䯈ˈV Ϟⱘ ⿃ ⒵䎇ϟ䗄 䋼ⱘ ˖(a)˖ ⿄ ˖!

! x y y x ,,(b)˖㒓 䋼˖!

! ! z y,z x,z y,x u s u s (c)˖䴲䋳 ˖ˈ0,!t y x Ϩ0

0 ! x x x,㢹 n 㓈㒓 ぎ䯈Ϟ НњϔϾ ⿃ˈ ⿄䖭Ͼぎ䯈Ў ⿃ぎ䯈DŽ⬅(a) (b)ˈ ⿃ Ѣ㄀Ѡ г 㒓 ⱘDŽ

⿃ぎ䯈Ё 䞣ⱘ䭓 ǃ 䞣䯈ⱘ 㾦ϢℷѸ ㄝὖ ˖䭓 (῵ 㣗 )˖ ! x x,||x 2||䎱⾏˖ 2

||y x y x ||),(d Cauchy ϡㄝ ˖ ! !d ! y y,x x,y x,2 㾦˖ !

! !

y y,x x,y x,2

2

xy

T cos ℷѸ˖ 0

! A y x,y x ℷѸ ˖ {}ˈ相n 1u ,...,u ¯®

­z ! j

i j i ij j i ,0,1G u ,u Fourier n

n 11u u x,...u u x,x x ! ! ,V Bessel’s ㄝ ˖ 2

222||||,!

! n 1u x,...u x,x x V Parseval’s ㄝ ˖ !! !! ! n u y y V ,,,,n 11u x,...u ,u x,y x y x Schmidt ℷѸ ˖ 㒓 ⱘ 䞣㒘ˈ Н˖

,...},{21v v B ⽣ ˖䅵ㅫ 㾚㾝Ёⱘ ⊩ , ㄀ ゴ˖ⶽ䰉ⱘ䜝 ℷѸ 㾦

119

,...

3,2,,1 ¦

!

! k k j 1k 1

j j j j k k 1u u ,u u ,v v u v u ℷѸ 䞣㒘ˈ Ϩ

}{,...u ,u 21 O k

span span },{}{k 21k 21v ,...,v ,v u ,...,u ,u ⃻⇣ぎ䯈R n

R n Ёⱘ( ) ⿃ НЎ˖

i

n

i i T y x ¦ ! 1

x y y x,ҹ ˈ ≵ ⡍ 䇈 ⱘ ϟˈ Ӏ 䅼䆎ⱘ⃻⇣ぎ䯈 ⬅Ϟ Нⱘ ⿃ぎ䯈R n DŽ

6.1.2ℷѸⶽ䰉

Н6.1.2 䰉Q ⒵䎇Q T Q ˙Q Q T =I ( ㄝӋ Q -1 = Q T )ˈ ⿄Q ЎℷѸⶽ䰉DŽℷѸⶽ䰉 ϟ䗄 䋼˖

(1) Q ЎℷѸⶽ䰉ӏ ϸϾ 䞣 ℷѸⱘϨӏ ϸϾ㸠 䞣г ԡℷѸⱘDŽ (2) Q ЎℷѸⶽ䰉 䞣ⱘ ⿃ϡ ˈ ˈ n R y x,!! y x,y x,Q Q DŽ(3) Q ЎℷѸⶽ䰉 䞣䯈ⱘ䎱⾏ϡ DŽ (4)ӏ ϾℷѸⶽ䰉ⱘ⿃ ℷѸⱘDŽ(5)ℷѸⶽ䰉ⱘ㸠 ㄝѢ1DŽ

r ℷѸⶽ䰉ⱘ㸠 ㄝѢ1ˈ ⿄ ℷ ⱘℷѸ䰉( 䕀)ˈ ⿄Ў䴲ℷ ℷѸ䰉( )DŽℷѸⶽ䰉ⱘ ԩ Н

2䰊ℷѸⶽ䰉Ҹ ℷѸ䰉ˈ

¸¸¹

·

¨¨©§ d c b a Q 22221d c b a ˈˈ22221d b c a cd

ab bd ac 0Ңⶹˈ 㾦122 b a T ˈՓ T T sin ,cos r b a DŽ T sin b ˈҸT T c ˈ

T T c sin a c ,cos b DŽ П Ӏ 䗝 ϔϾ㾦T Փ

T

T sin ,cos b a

(6.1.1)

⫼ˈ 122 c a T sin r c DŽ ˈ⬅0 bd ac ˈ T cos r d (sin 0z T )DŽ䖭䇈 ˈ 0z sin T ˈ

T sin c ˈT cos d T sin c ˈT

cos d ҹˈ 0sin z T ˈQ ҙ ϟ䗄ϸ⾡㉏ ˖

¸¸¹·¨¨©§ T T T T

cos sin sin cos ˈ¨¨ (6.1.2)

¸¸¹

·

©§ T T T T

cos sin sin cos ⽣ ˖䅵ㅫ 㾚㾝Ёⱘ ⊩ , ㄀ ゴ˖ⶽ䰉ⱘ䜝 ℷѸ 㾦 120