基礎数学の公式

جૅ਺ֶͷެࣜ ϕΠζ౷ܭͷཧ࿦ͱํ๏ษڧձ

సஔߦྻ τϨʔε ߦྻࣜ

సஔߦྻɺϕΫτϧ Ұൠʹ ߦྻ ʹ͍ͭͯɺ ͦͷసஔߦྻ ͱ͸ ߦྻͰ ͷ͜ͱͰ͋Δɻ k ×
d A = (Aij ) AT d × k A = (Aji ) ࣍ݩϕΫτϧ Λ ߦྻͱߟ͑Δɻ͜ΕΛॎϕΫτϧͱ͍͏ɻ ·ͨɺ͜ͷͱ͖ԣϕΫτϧ ͸ ߦྻͰ͋Δɻ d v d × 1 vT 1 × d

τϨʔε ߦྻ ͷτϨʔεΛ d × d A tr(A) =
d ∑ i=1 Aii ͱॻ͘ɻ tr(AB) = tr(BA) ͕੒Γཱͭɻ ߦྻ ʹ͍ͭͯҰൠʹ ͕ͩɺҰൠʹ d × d A, B AB ≠ BA ର֯੒෼ͷ࿨

಺ੵ Λ ࣍ݩͷϕΫτϧͱ͢Δͱ͖ɺͦͷ಺ੵΛ u, v d u ⋅ v =
d ∑ i=1 ui vi ͱॻ͘ɻ

ϊϧϜ ϕΫτϧ ͷϊϧϜΛ Ͱද͢ɻ Ͱ͋Δɻ u ∥u∥ ∥u∥ = u
⋅ u u ⋅ v = uTv = tr(v ⋅ uT) ͕੒Γཱͭɻ͜ΕΑΓ (u ⋅ v)2 = vTuuTv = tr(uuTvvT) ͕ಘΒΕΔɻ·ͨ u ⋅ Av = (ATu) ⋅ v = tr(AvuT) ͕੒Γཱͭɻ

ߦྻࣜ ߦྻ ͷߦྻࣜΛ d × d A det(A) = ∑
σ sgn(σ)A1σ(1) A2σ(2) ⋯Adσ(d) ͱॻ͘ɻ ͜͜Ͱ ͸ཁૉͷ਺͕ ݸͷू߹͔Βࣗ෼ࣗ਎΁ͷશ୯ࣹ ஔ׵ Λද͢ɻ ஔ׵ͷݸ਺͸શ෦Ͱ ݸͰ͋Δ͕ɺ ͸ஔ׵શମͷू߹ʹର͢Δ࿨Ͱ͋Γɺ σ d d! ∑ σ ྫ ͸حஔ׵ͷͱ͖ ɺۮஔ׵ͷͱ͖ Ͱ͋Δɻ sgn(σ) −1 1 IUUQTPHVFNPODPNTUVEZMJOFBSBMHFCSBEFUXIBU

ߦྻࣜ ߦྻ ʹ͍ͭͯ d × d A, B det(AB) =
det(A) det(B) ͕੒Γཱͭɻ

ରশߦྻ ݻ༗஋ ਖ਼ఆ஋ߦྻ

ਖ਼ଇɾٯߦྻɾରশߦྻɾ௚ߦߦྻ ߦྻ ͕Մٯ͋Δ͍͸ਖ਼ଇͰ͋Δͱ͸ ͕୯ҐߦྻͱͳΔΑ͏ͳߦྻ ͕ଘࡏ͢Δ͜ͱͰ͋Δɻ d × d A A−1A
A−1 ͕ՄٯͰ͋Δͱ͖ Λ ͷٯߦྻͱ͍͏ɻ A A−1 A ࣮਺Λཁૉʹ࣋ͭ ߦྻ ͕ରশߦྻͰ͋Δͱ͸ɺ ͕੒Γཱͭ͜ͱͰ͋Δɻ d × d A = (Aij ) A = AT ࣮਺Λཁૉʹ࣋ͭ ߦྻ ͕௚ߦߦྻͰ͋Δͱ͸ɺ ͕୯ҐߦྻͰ͋Δ͜ͱͰ͋Δɻ d × d R = (Rij ) RTR

ର֯ߦྻɾର֯Խ ߦྻ ͕ର֯ߦྻͰ͋Δͱ͸ ͕੒Γཱͭ͜ͱͰ͋Δɻ d × d A i ≠
j ⇒ Aij = 0 ೚ҙͷରশߦྻ ʹରͯ͠ Λର֯ߦྻʹ͢ΔΑ͏ͳ௚ߦߦྻ ͕ଘࡏ͢Δɻ ͢ͳΘͪ A R−1AR R ͱग़དྷΔɻ͜ͷͱ͖ର֯ߦྻ ΛٻΊΔ͜ͱΛʮ Λର֯Խ͢Δʯͱ͍͏ɻ R−1AR A

δϣϧμϯඪ४ܗ ର֯ߦྻͰͳ͍ߦྻ͸Ұൠతʹ͸ର֯Խग़དྷΔͱ͸ݶΒͳ͍͕ɺ ՄٯߦྻΛ༻͍ͯδϣϧμϯඪ४ܗʹ͢Δ͜ͱ͕ग़དྷΔɻ δϣϧμϯࡉ๔ δϣϧμϯඪ४ܗ

ݻ༗஋ɾਖ਼ఆ஋ߦྻ ߦྻ ʹ͍ͭͯɺෳૉ਺ ͱෳૉ਺Λཁૉͱ͢ΔϕΫτϧ ͕ ଘࡏͯ͠ d × d A
λ v ≠ 0 ͕੒Γཱͭͱ͖ɺ Λ ͷݻ༗஋ͱ͍͍ɺ Λ ͷݻ༗ϕΫτϧͱ͍͏ɻ ରশߦྻ ͷݻ༗஋͕શͯ ΑΓେ͖͍ͱ͖ɺ Λਖ਼ఆ஋ߦྻͰ͋Δͱ͍͏ɻ λ A v A A 0 A Av = λv ͕ਖ਼ఆ஋ߦྻͰ͋Ε͹ೋ࣍ࣜʹ͍ͭͯฏํ׬੒͕ग़དྷΔɻ A 1 2 (u ⋅ Au) − u ⋅ v = 1 2 ∥A1/2(u − A−1v)∥2 − 1 2 ∥A−1/2v∥2

ੵ෼ެࣜ

ਖ਼ن෼෍ ͱ͠ɺ Λ ͷਖ਼ఆ஋ߦྻͰ͋Δͱ͢Δɻ͜ͷͱ͖ w ∈ ℝd A d ×
d ∫ exp(− n 2 w ⋅ A−1w)dw = (2π)d/2 det(A)1/2 nd/2 ฏۉ͕ Ͱ෼ࢄڞ෼ࢄߦྻ͕ ͷਖ਼ن෼෍͸ a ∈ ℝd A (a, A) = p(w) = 1 (2π)d/2 det(A)1/2 exp(− 1 2 (w − a) ⋅ A−1(w − a)) ͱ͍͏ࣜͰද͞ΕΔɻ

ਖ਼ن෼෍ ͜ͷͱ͖ ∫ wp(w)dw = a Ͱ͋Γɺ೚ҙͷ ߦྻ ʹ͍ͭͯ d
× d B ∫ (w − a) ⋅ B(w − a)p(w)dw = tr(BA) ͷ֬཰෼෍͸ Ͱ͋Δɻ (Ba + b, BABT) Ͱ͋Δɻ Λ ͷՄٯߦྻͱ͢Δɻ ֬཰ม਺ ͷ֬཰෼෍͕ Ͱ͋Δͱ͖֬཰ม਺ B d × d X (a, A) Y = BX + b

ฏۉ஋ͷఆཧ

ଟม਺͔ΒͳΔؔ਺ͷඍ෼ͷه๏ Λඇෛͷ੔਺ͱ͢Δɻଟॏࢦ਺ ʹ͍ͭͯ࣍ͷΑ͏ʹఆΊΔɻ k1 , …, kd k = (k1
, …, kd ) |k| = k1 + k2 + ⋯ + kd , k! = (k1 )!(k2 )!⋯(kd )! ͷ ΛؚΉ։ू߹্Ͱఆٛ͞Εͨ ճ࿈ଓඍ෼Մೳͳؔ਺ ʹ͍ͭͯ ͷͱ͖ ℝd w = w0 (r + 1) g(w) |k| ≤ r + 1 ∂kg ∂wk (w) = ( d ∏ j=1 ∂kj ∂wkj j )g(w), ͱఆٛ͢Δɻ ͜ΕΒ͸ଟม਺ͷؔ਺ͷඍ෼ʹ͓͍ͯҰൠతʹར༻͞ΕΔه๏Ͱ͋Δɻ (w − w0 )k = d ∏ j=1 (wj − (w0 )j )kj

ฏۉ஋ͷఆཧ ೚ҙͷ ʹରͯ͠ɺ ʹΑΔఆ·Δ ͕ଘࡏͯ͠ w w w* g(w) =
∑ |k|≤r ∂kg ∂wk (w0 ) (w − w0 )k k! + ∑ |k|=r+1 ∂kg ∂wk (w*) (w − w0 )k k! ͕੒ཱ͢Δɻ ͜Ε͕ฏۉ஋ͷఆཧͰ͋Γɺؔ਺ͷมԽͷฏۉతͳڍಈʹ͍ͭͯͷఆཧͰ͋Δɻ ·ͨ͜͜Ͱ ͸͋Δ ͕ଘࡏͯ͠ w* 0 < θ < 1 w* = w0 + θ(w − w0 ) ͱද͞ΕΔɻ ಛʹ Ͱ͋Δ͔Βɺ ͷͱ͖ ͕੒Γཱͭɻ ∥w* − w0 ∥ ≤ ∥w − w0 ∥ w → w0 w* → w0

基礎数学の公式

基礎数学の公式

Koga Kobayashi

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جૅ਺ֶͷެࣜ ϕΠζ౷ܭͷཧ࿦ͱํ๏ษڧձ

సஔߦྻ τϨʔε ߦྻࣜ

సஔߦྻɺϕΫτϧ Ұൠʹ ߦྻ ʹ͍ͭͯɺ ͦͷసஔߦྻ ͱ͸ ߦྻͰ ͷ͜ͱͰ͋Δɻ k ×

τϨʔε ߦྻ ͷτϨʔεΛ d × d A tr(A) =

಺ੵ Λ ࣍ݩͷϕΫτϧͱ͢Δͱ͖ɺͦͷ಺ੵΛ u, v d u ⋅ v =

ϊϧϜ ϕΫτϧ ͷϊϧϜΛ Ͱද͢ɻ Ͱ͋Δɻ u ∥u∥ ∥u∥ = u

ߦྻࣜ ߦྻ ͷߦྻࣜΛ d × d A det(A) = ∑

ߦྻࣜ ߦྻ ʹ͍ͭͯ d × d A, B det(AB) =

ରশߦྻ ݻ༗஋ ਖ਼ఆ஋ߦྻ

ਖ਼ଇɾٯߦྻɾରশߦྻɾ௚ߦߦྻ ߦྻ ͕Մٯ͋Δ͍͸ਖ਼ଇͰ͋Δͱ͸ ͕୯ҐߦྻͱͳΔΑ͏ͳߦྻ ͕ଘࡏ͢Δ͜ͱͰ͋Δɻ d × d A A−1A

ର֯ߦྻɾର֯Խ ߦྻ ͕ର֯ߦྻͰ͋Δͱ͸ ͕੒Γཱͭ͜ͱͰ͋Δɻ d × d A i ≠

δϣϧμϯඪ४ܗ ର֯ߦྻͰͳ͍ߦྻ͸Ұൠతʹ͸ର֯Խग़དྷΔͱ͸ݶΒͳ͍͕ɺ ՄٯߦྻΛ༻͍ͯδϣϧμϯඪ४ܗʹ͢Δ͜ͱ͕ग़དྷΔɻ δϣϧμϯࡉ๔ δϣϧμϯඪ४ܗ

ݻ༗஋ɾਖ਼ఆ஋ߦྻ ߦྻ ʹ͍ͭͯɺෳૉ਺ ͱෳૉ਺Λཁૉͱ͢ΔϕΫτϧ ͕ ଘࡏͯ͠ d × d A

ੵ෼ެࣜ

ਖ਼ن෼෍ ͱ͠ɺ Λ ͷਖ਼ఆ஋ߦྻͰ͋Δͱ͢Δɻ͜ͷͱ͖ w ∈ ℝd A d ×

ਖ਼ن෼෍ ͜ͷͱ͖ ∫ wp(w)dw = a Ͱ͋Γɺ೚ҙͷ ߦྻ ʹ͍ͭͯ d

ฏۉ஋ͷఆཧ

ଟม਺͔ΒͳΔؔ਺ͷඍ෼ͷه๏ Λඇෛͷ੔਺ͱ͢Δɻଟॏࢦ਺ ʹ͍ͭͯ࣍ͷΑ͏ʹఆΊΔɻ k1 , …, kd k = (k1

ฏۉ஋ͷఆཧ ೚ҙͷ ʹରͯ͠ɺ ʹΑΔఆ·Δ ͕ଘࡏͯ͠ w w w* g(w) =