190306_2章_ver2

2019/03/06 ʮਂ૚ֶशʯୈ2ষ 2-1~2-8

εΧϥʔʢscalarsʣ ୯Ұͷ਺ ͲΜͳछྨͷ਺ͳͷ͔໌ࣔ ▸ ࣮਺஋ͷεΧϥʔ ▸ ࣗવ਺ͷεΧϥʔ  2 s ∈
ℝ n ∈ ℕ

ϕΫτϧʢvectorsʣ ਺஋ͷ഑ྻ ▸ ্ۭؒͷ఺Λද͍ͯ͠ΔͱΈͳͤΔ ▸ ͦΕͧΕͷཁૉ͸֤࣠ͷ࠲ඪ ཁૉͷू߹ʹΠϯσοΫεΛৼΔ৔߹ ▸ ΠϯσοΫεΛؚΉू߹Λఆٛ ɹɹɹɹɹɹΛର৅ͱ͢Δ࣌ɼू߹ɹɹɹɹɹΛఆٛˠ
▸ ه߸͸͋Δू߹ͷิू߹ ɹɹɹɹ͸ɹͷ͏ͪɹҎ֎ͷશཁૉΛؚΉϕΫτϧ ɹɹɹɹ͸ɹͷ͏ͪɹɹɹɹҎ֎ͷશཁૉΛؚΉϕΫτϧ 3 x1 , x3 , x6 S = {1,3,6} x−1 x x1 x−S x x1 , x3 , x6 xS

ߦྻʢmatricesʣ ͭͷΠϯσοΫεͰࣔ͞ΕΔ਺஋഑ྻ ▸ ࣮਺஋ߦྻɹͷߴ͕͞ɹͰ෯͕ ▸ ɿ͸͢΂ͯͷཁૉΛࢦఆɹ ▸ ɹ͸ɹͷਨ௚࠲ඪɹʹ͓͚Δਫฏͳஅ໘ʢɹ൪໨ͷߦʣ 4 A
m n A ∈ ℝm×n A = [ A1,1 A1,2 A2,1 A2,2 ] (m = n = 2) Ai,: A i i

ςϯιϧʢUFOTPSTʣ ͭҎ্ͷ࣠Λ࣋ͭ഑ྻ ͭͷ࣠ͷೖྗʹରͯ͠  ग़ྗ͕͍͔ͭ͘ͷ࣠ʹग़Δ͜ͱΛ਺ࣜͰද͢ ྫ͑͹෺ʹରͯ͋͠Δํ޲͔ΒྗΛՃ͑Δ ▸ ྗ͕ՃΘͬͨͷ͸ํ޲ͳͷʹର͠  ෺͕มܗ͢ΔมԽྔ͸ํ޲ٻΊΔඞཁ 5

సஔʢtransposeʣ ߦྻʹؔ͢Δԋࢉͷͭ ▸ ߦྻͷର֯ઢʹର͢Δڸ૾ ϕΫτϧͷసஔ ▸ ϕΫτϧ͸ྻͷߦྻˠసஔ͸ߦͷߦྻ εΧϥʔͷసஔ ▸ ཁૉΛ͚ͭͩ࣋ͭߦྻˠ
6 A = A1,1 A1,2 A2,1 A2,2 A3,1 A3,2 㱺 AT = [ A1,1 A2,1 A3,1 A1,1 A2,2 A3,2 ] (AT)i,j = Aj,i aT = a

ߦྻͷՃࢉ ▸ ରԠ͢ΔཁૉΛ଍͢  ɹɹɹɹɹɼ ߦྻ΁ͷεΧϥʔͷՃࢉɼ৐ࢉ ▸ ֤ཁૉʹରͯ͠ԋࢉΛߦ͏  ɹɹɹɹɹɹɼ 7 సஔʢtransposeʣ
C = A + B Ci,j = Ai,j + Bi,j D = a ⋅ B + c Di,j = a ⋅ Bi,j + c ਂ૚ֶशͷจ຺Ͱ͸ʜ ɹɹɹɹʢϕΫτϧɹΛߦྻͷ֤ߦʹՃࢉʣΛڐ༰ ɹ‎৽ͨͳߦྻͷఆ͕ٛෆཁ C = A + b b ϒϩʔυΩϟεςΟϯά

ߦྻͱϕΫτϧͷ৐ࢉ ߦྻɹͱɹͷੵ͕ߦྻ ▸ ɹͷྻͷ਺ɹɹͷߦͷ਺ ▸ ɹ͕ɹɹɹɼɹ͕ɹɹɹͳΒɼɹ͸ ͭͷϕΫτϧɹɹͷυοτੵ ▸ ߦྻͷੵɹɹͰٻ·Δ ▸
ɹɹɹ͸ɹͷɹߦͱɹͷɹྻͷ಺ੵͱΈͳͤΔ 8 A B C A = B A m × n B n × p C m × p C = AB Ci,j = ∑ k Ai,k Bk,j x, y xTy C = AB A i B j

ߦྻͷϕΫτϧͷ৐ࢉ ɹɹɹɹ͸ඞͣ͠΋੒Γཱͨͳ͍ɹ ▸ ͭͷϕΫτϧؒͷ಺ੵ͸Մ׵ ߦྻͷੵͷసஔ 9 AB = BA xTy
= yTx (1) (ATB) = BTA (2) ɹͷ஋͸εΧϥʔͰͦΕࣗ਎ͷసஔʹ౳͍͠ ɹ‎ΑΓ xTy = (xTy)T = yTx (1) (2)

୯Ґߦྻͱٯߦྻ ୯Ґߦྻ ▸ ɹ࣍ݩͷ୯Ґߦྻ ▸ ೚ҙͷϕΫτϧʹֻ͚ͯ΋஋Λม͑ͳ͍ߦྻ ٯߦྻ ▸ ɹɹͱද͞Εɼఆٛ͸ 10
n In In x = x I3 = [ 1 0 0 0 1 0 0 0 1 ] ओର֯ઢʹԊͬͨ੒෼͕ A−1 A−1A = In

ٯߦྻ๏ ઢܗํఔࣜɹɹɹɹΛٯߦྻΛ༻͍ͯղ͘ 11 Ax = b A−1A = In ΑΓ
A−1Ax = A−1b Ax = b In x = A−1b x = A−1b ɹɹ͸ཧ࿦తͳखஈͱͯ͠༗༻ σδλϧͳίϯϐϡʔλͰ͸ɹɹͷਫ਼౓͕ݶఆ͞ΕΔ ɹ‎ͷ஋Λར༻ A−1 A−1 b

ɹɹɹɹΛٯߦྻΛ༻͍ͯղ͚Δ ˣ ํఔࣜͷղ͸͚̍ͭͩղΛ࣋ͭ ઢܗ݁߹ ɹɹɹ ▸ ͍͔ͭ͘ͷϕΫτϧ͔ΒͳΔू߹ͷઢܗ݁߹ɹɹɹɹɹɹ 12 Ax =
b ɹͷྻ͕ɼݪ఺͔Βͷํ޲Λ͍ࣔͯͯ͠ ɹʹ౸ୡ͢Δܦ࿏ͷ਺ΛܾΊΔ΋ͷͩͱߟ͑Δ ɹͷ֤ཁૉ͸ͦΕͧΕͷํ޲ʹਐΉڑ཭ A b x Ax = ∑ i xi A:,j xi ͸ྻɹͷํ޲ʹਐΉڑ཭ i ∑ i ci v(i)

ઢܗैଐͱுΔۭؒ ுΔۭؒ ▸ ݩͷϕΫτϧͷઢܗ݁߹ʹΑͬͯಘΒΕΔ  ఺͔ΒͳΔू߹ ▸ ɹɹɹʹղ͕͋Δ͔͸ɼɹ  ɹ͕ɹͷྻ͕షΔۭؒʹؚ·ΕΔ͔Ͳ͏͔ ઢܗैଐʢ৑௕ੑʣ ▸
ྫ͑͹ºߦྻʹ͓͍ͯɼ  ର৅ͱͳΔɹ͸࣍ݩ͕ͩɼɹ͸࣍ݩ  ɹˠ͕ுΔۭؒ಺ͷ࣍ݩฏ໘಺Ͱ͔͠ಈ͚ͳ͍ ઢܗಠཱ ▸ ͲͷϕΫτϧ΋ଞͷϕΫτϧͱઢܗ݁߹Ͱͳ͍ 13 Ax = b A b A b x

ߦྻ͕ٯߦྻΛ࣋ͭ৚݅ ▸ ਖ਼ํߦྻ͕ਖ਼ଇͰͳ͚Ε͹ͳΒͳ͍  ˠྻϕΫτϧ͸ޓ͍ʹઢܗಠཱ  ͷྻϕΫτϧΛฒ΂ͨϕΫτϧΛɹɹɹɹɹɹɹɹɹͱ͢Δͱ ɹ͸ਖ਼ଇߦྻ͔ͩΒ ٯߦྻͷ৚݅ 14 A A−1A
= AA−1 = I ͜ͷͱ͖ Ax = 0 Λຬͨ͢ x ͸ x = 0 ͷΈ A A = [a1 a2 . . . an] x1 a1 + x2 a2 + . . . + xn an = 0 ˠ x1 = x2 = . . . = xn = 0 ΑͬͯϕΫτϧͷ૊ɹɹɹɹɹɹ͸ઢܗಠཱͰ͋Δ a1 , a2 , . . . , an

ϊϧϜʢnormʣ ϕΫτϧͷେ͖͞ΛٻΊΔؔ਺ ϢʔΫϦουϊϧϜ ▸ ɹɹɹͰ͋ΔɹϊϧϜ ▸ ݪ఺͔ΒɹͰࣔ͞ΕΔ఺·ͰͷϢʔΫϦουڑ཭ ▸ ϕΫτϧͷେ͖͞ɹϊϧϜͷ৐ 15
∥x∥p = (∑ i |xi |p )1 p (p ∈ ℝ, p ≥ 1) p = 2 Lp L2 x = L2 = xTx

ɹϊϧϜ ▸ θϩͷཁૉͱθϩҎ֎ͷཁૉͷ۠ผ͕ॏཁ ▸ ཁૉ͕͔Β཭Εͨ෼͚ͩ૿Ճ͢ΔɹϊϧϜ ▸ ϕΫτϧ಺ͷඇཁૉͷ਺ ࠷େ஋ϊϧϜ ▸ ϕΫτϧͷதͰઈର஋͕࠷େͷཁૉͷઈର஋
16 ϊϧϜʢnormʣ L1 ∥x∥1 = ∑ i |xi | L1 L∞ ∥x∥∞ = max i |xi |

ಛผͳߦྻͱϕΫτϧ ର֯ߦྻ ▸ ओର֯੒෼ͷΈඇ੒෼Ͱ͋Δߦྻ ▸ ୯Ґߦྻ͸͢΂ͯͷର֯੒෼͕ ▸ ֻ͚ࢉͷܭࢉޮ཰͕ྑ͍ ରশߦྻ ▸
సஔߦྻ͕ݩͷߦྻͱಉ͡ 17 Ai,j = Aj,i diag(v) ର֯ߦྻɹɹɹɹͱɹΛֻ͚߹ΘͤΔͱ͖  ֤ཁૉɹΛɹഒ͢Δ͚ͩ x xi vi

ಛผͳߦྻͱϕΫτϧ ୯ҐϕΫτϧ ▸ ௕͞ʢ୯ҐϊϧϜʣ͕ͷϕΫτϧ ਖ਼ن௚ަ ▸ ϕΫτϧɹͱϕΫτϧɹ͕ɹɹͷͱ͖௚ަ ▸ ޓ͍ʹ౓ͷ֯Λ੒͍ͯ͠Δ ▸
୯ҐϊϧϜ΋࣋ͭͱ͖ɼਖ਼ن௚ަ ௚ަߦྻ ▸ ਖ਼ํߦྻ͔ͭਖ਼ن௚ަ 18 xTy x y ATA = AAT = I A−1 = AT ˣ

ݻ༗஋෼ղ ߦྻ෼ղͷͭ ▸ ੒෼ͷ഑ྻ͔Β͸ݟ͑ͳ͍ػೳతੑ࣭Λ஌Δ ߦྻΛݻ༗ϕΫτϧͱݻ༗஋ͷू߹ʹ෼ղ ਖ਼ํߦྻɹͷݻ༗ϕΫτϧ ▸ ɹΛֻ͚ͨͱ͖εΧϥʔഒͷΈɼ޲͖͸มԽ͠ͳ͍ ▸ ͜ͷ࣌ͷεΧϥʔɹ͕ݻ༗஋
19 Av = λv A v v λ A = Vdiag(λ)V−1 ݻ༗ϕΫτϧΛ࿈݁ͯ͠ɹͱ͢Δͱ V ݻ༗஋෼ղɿ

ݻ༗஋෼ղ ݻ༗஋෼ղͷྲྀΕ ݻ༗ϕΫτϧɹɹɹΛฒ΂ͯͰ͖Δߦྻ ɹ 20 v1 , v2 Q [Av1
Av2] = [λv1 λv2] A [v1 v2] = [λv1 λv2] [ λ1 0 0 λ2 ] Q = [v1 v2] ͔Β A−1AQ = [ λ1 0 0 λ2 ] AQ = Q [ λ1 0 0 λ2 ]

ಛҟ஋෼ղ ݻ༗஋෼ղ͸ਖ਼ํߦྻͰͷΈఆٛ͞ΕΔ ɹݻ༗ϕΫτϧͷߦྻɹͱݻ༗஋ɹͷϕΫτϧ͔Β ಛҟ஋෼ղͰ͸ͭͷߦྻͰɹΛද͢ ɹ͸ඞͣ͠΋ਖ਼ํߦྻͰͳ͍ ɹ͸ର֯ߦྻͰର֯੒෼Λಛҟ஋ͱ͍͏ 21 A = Vdiag(λ)V−1
λ V ݻ༗஋෼ղɿ A A = UDV−T U = m × m A = m × n V = n × n D = m × n D D

190306_2章_ver2

190306_2章_ver2

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2019/03/06 ʮਂ૚ֶशʯୈ2ষ 2-1~2-8

εΧϥʔʢscalarsʣ ୯Ұͷ਺ ͲΜͳछྨͷ਺ͳͷ͔໌ࣔ ▸ ࣮਺஋ͷεΧϥʔ ▸ ࣗવ਺ͷεΧϥʔ  2 s ∈

ϕΫτϧʢvectorsʣ ਺஋ͷ഑ྻ ▸ ্ۭؒͷ఺Λද͍ͯ͠ΔͱΈͳͤΔ ▸ ͦΕͧΕͷཁૉ͸֤࣠ͷ࠲ඪ ཁૉͷू߹ʹΠϯσοΫεΛৼΔ৔߹ ▸ ΠϯσοΫεΛؚΉू߹Λఆٛ ɹɹɹɹɹɹΛର৅ͱ͢Δ࣌ɼू߹ɹɹɹɹɹΛఆٛˠ

ߦྻʢmatricesʣ ͭͷΠϯσοΫεͰࣔ͞ΕΔ਺஋഑ྻ ▸ ࣮਺஋ߦྻɹͷߴ͕͞ɹͰ෯͕ ▸ ɿ͸͢΂ͯͷཁૉΛࢦఆɹ ▸ ɹ͸ɹͷਨ௚࠲ඪɹʹ͓͚Δਫฏͳஅ໘ʢɹ൪໨ͷߦʣ 4 A

ςϯιϧʢUFOTPSTʣ ͭҎ্ͷ࣠Λ࣋ͭ഑ྻ ͭͷ࣠ͷೖྗʹରͯ͠  ग़ྗ͕͍͔ͭ͘ͷ࣠ʹग़Δ͜ͱΛ਺ࣜͰද͢ ྫ͑͹෺ʹରͯ͋͠Δํ޲͔ΒྗΛՃ͑Δ ▸ ྗ͕ՃΘͬͨͷ͸ํ޲ͳͷʹର͠  ෺͕มܗ͢ΔมԽྔ͸ํ޲ٻΊΔඞཁ 5

సஔʢtransposeʣ ߦྻʹؔ͢Δԋࢉͷͭ ▸ ߦྻͷର֯ઢʹର͢Δڸ૾ ϕΫτϧͷసஔ ▸ ϕΫτϧ͸ྻͷߦྻˠసஔ͸ߦͷߦྻ εΧϥʔͷసஔ ▸ ཁૉΛ͚ͭͩ࣋ͭߦྻˠ

ߦྻͷՃࢉ ▸ ରԠ͢ΔཁૉΛ଍͢  ɹɹɹɹɹɼ ߦྻ΁ͷεΧϥʔͷՃࢉɼ৐ࢉ ▸ ֤ཁૉʹରͯ͠ԋࢉΛߦ͏  ɹɹɹɹɹɹɼ 7 సஔʢtransposeʣ

ߦྻͱϕΫτϧͷ৐ࢉ ߦྻɹͱɹͷੵ͕ߦྻ ▸ ɹͷྻͷ਺ɹɹͷߦͷ਺ ▸ ɹ͕ɹɹɹɼɹ͕ɹɹɹͳΒɼɹ͸ ͭͷϕΫτϧɹɹͷυοτੵ ▸ ߦྻͷੵɹɹͰٻ·Δ ▸

ߦྻͷϕΫτϧͷ৐ࢉ ɹɹɹɹ͸ඞͣ͠΋੒Γཱͨͳ͍ɹ ▸ ͭͷϕΫτϧؒͷ಺ੵ͸Մ׵ ߦྻͷੵͷసஔ 9 AB = BA xTy

୯Ґߦྻͱٯߦྻ ୯Ґߦྻ ▸ ɹ࣍ݩͷ୯Ґߦྻ ▸ ೚ҙͷϕΫτϧʹֻ͚ͯ΋஋Λม͑ͳ͍ߦྻ ٯߦྻ ▸ ɹɹͱද͞Εɼఆٛ͸ 10

ٯߦྻ๏ ઢܗํఔࣜɹɹɹɹΛٯߦྻΛ༻͍ͯղ͘ 11 Ax = b A−1A = In ΑΓ

ɹɹɹɹΛٯߦྻΛ༻͍ͯղ͚Δ ˣ ํఔࣜͷղ͸͚̍ͭͩղΛ࣋ͭ ઢܗ݁߹ ɹɹɹ ▸ ͍͔ͭ͘ͷϕΫτϧ͔ΒͳΔू߹ͷઢܗ݁߹ɹɹɹɹɹɹ 12 Ax =

ઢܗैଐͱுΔۭؒ ுΔۭؒ ▸ ݩͷϕΫτϧͷઢܗ݁߹ʹΑͬͯಘΒΕΔ  ఺͔ΒͳΔू߹ ▸ ɹɹɹʹղ͕͋Δ͔͸ɼɹ  ɹ͕ɹͷྻ͕షΔۭؒʹؚ·ΕΔ͔Ͳ͏͔ ઢܗैଐʢ৑௕ੑʣ ▸

ߦྻ͕ٯߦྻΛ࣋ͭ৚݅ ▸ ਖ਼ํߦྻ͕ਖ਼ଇͰͳ͚Ε͹ͳΒͳ͍  ˠྻϕΫτϧ͸ޓ͍ʹઢܗಠཱ  ͷྻϕΫτϧΛฒ΂ͨϕΫτϧΛɹɹɹɹɹɹɹɹɹͱ͢Δͱ ɹ͸ਖ਼ଇߦྻ͔ͩΒ ٯߦྻͷ৚݅ 14 A A−1A

ϊϧϜʢnormʣ ϕΫτϧͷେ͖͞ΛٻΊΔؔ਺ ϢʔΫϦουϊϧϜ ▸ ɹɹɹͰ͋ΔɹϊϧϜ ▸ ݪ఺͔ΒɹͰࣔ͞ΕΔ఺·ͰͷϢʔΫϦουڑ཭ ▸ ϕΫτϧͷେ͖͞ɹϊϧϜͷ৐ 15

ɹϊϧϜ ▸ θϩͷཁૉͱθϩҎ֎ͷཁૉͷ۠ผ͕ॏཁ ▸ ཁૉ͕͔Β཭Εͨ෼͚ͩ૿Ճ͢ΔɹϊϧϜ ▸ ϕΫτϧ಺ͷඇཁૉͷ਺ ࠷େ஋ϊϧϜ ▸ ϕΫτϧͷதͰઈର஋͕࠷େͷཁૉͷઈର஋

ಛผͳߦྻͱϕΫτϧ ର֯ߦྻ ▸ ओର֯੒෼ͷΈඇ੒෼Ͱ͋Δߦྻ ▸ ୯Ґߦྻ͸͢΂ͯͷର֯੒෼͕ ▸ ֻ͚ࢉͷܭࢉޮ཰͕ྑ͍ ରশߦྻ ▸

ಛผͳߦྻͱϕΫτϧ ୯ҐϕΫτϧ ▸ ௕͞ʢ୯ҐϊϧϜʣ͕ͷϕΫτϧ ਖ਼ن௚ަ ▸ ϕΫτϧɹͱϕΫτϧɹ͕ɹɹͷͱ͖௚ަ ▸ ޓ͍ʹ౓ͷ֯Λ੒͍ͯ͠Δ ▸

ݻ༗஋෼ղ ߦྻ෼ղͷͭ ▸ ੒෼ͷ഑ྻ͔Β͸ݟ͑ͳ͍ػೳతੑ࣭Λ஌Δ ߦྻΛݻ༗ϕΫτϧͱݻ༗஋ͷू߹ʹ෼ղ ਖ਼ํߦྻɹͷݻ༗ϕΫτϧ ▸ ɹΛֻ͚ͨͱ͖εΧϥʔഒͷΈɼ޲͖͸มԽ͠ͳ͍ ▸ ͜ͷ࣌ͷεΧϥʔɹ͕ݻ༗஋

ݻ༗஋෼ղ ݻ༗஋෼ղͷྲྀΕ ݻ༗ϕΫτϧɹɹɹΛฒ΂ͯͰ͖Δߦྻ ɹ 20 v1 , v2 Q [Av1

ಛҟ஋෼ղ ݻ༗஋෼ղ͸ਖ਼ํߦྻͰͷΈఆٛ͞ΕΔ ɹݻ༗ϕΫτϧͷߦྻɹͱݻ༗஋ɹͷϕΫτϧ͔Β ಛҟ஋෼ղͰ͸ͭͷߦྻͰɹΛද͢ ɹ͸ඞͣ͠΋ਖ਼ํߦྻͰͳ͍ ɹ͸ର֯ߦྻͰର֯੒෼Λಛҟ஋ͱ͍͏ 21 A = Vdiag(λ)V−1