linear algebra

第２回 TensorFlowと機械学習に必要な数学を基礎から学ぶ会資料 線形代数

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1. ઢܗ਺ֶ Linear algebra ઙ઒৳Ұ ฏ੒ 28 ೥ 10 ݄ 12 ೔ 1 Vector ͍͔ͭ͘ͷ਺஋Λ·ͱΊͯදݱ͢Δํ๏Ͱ͋Δ. x = (x1 , x2 , . . . , xn ) ͜Ε ʹରͯ͠, ී௨ͷҙຯͰͷ਺஋ΛεΧϥ scalor ͱ͍͏. ߦϕΫτϧ raw vectorͱ͸ n ݸͷ਺஋͕ԣʹฒΜͩ΋ͷͰɼ a = (a1 , a2 , . . . , an ) ͱදه͢ΔɽྻϕΫτϧ column vector ͱ͸ n ݸͷ਺஋͕ॎʹฒΜͩ΋Ͱ͋ Γɼb =       b1 b2 . . . bn       ͓Αͼ bt = (b1 , b2 , . . . , bn ) ͱϕΫτϧͷݞʹ t ΍ ′ Λͭ ͚ͯදه͢Δ͜ͱ΋͋Δɽ 1.1 ϊϧϜ ϕΫτϧͷϊϧϜ norm ͋Δ͍͸ϢʔΫϦουϊϧϜͱ͸ϕΫτϧͷ௕͞ ͷ͜ͱͰ͋Γɼ|a| = √ a′a = ∑ i a2 i ͱදه͢Δɽ|x| = 1 ͳΒ͹ਖ਼ଇԽ͞Ε ͨϕΫτϧͱ͍͏ɽ 1.2 ಺ੵ ಉ࣍͡ݩΛ΋ͭ 2 ͭͷϕΫτϧͷ಺ੵ inner product Λ atb ·ͨ͸ (a · b) ͱදه͢Δɽ͜Ε͸ (a · b) = |a| |b| cos θ = n ∑ i=1 ai bi Ͱ͋Δɽ2 ͭͷϕΫτϧ ͳ֯͢ θ ͸ cos θ = (a · b) |a| |b| (1) Ͱ͋Δɽ಺ੵ͸ 2 ͭͷϕΫτϧͷڞઢੑɼ͢ͳΘͪฏߦͷ౓߹ͷई౓ͱͳΓɼ ಛʹ (a · b) = 0 ͳΒ͹ͦΕΒ 2 ຊͷϕΫτϧ͸௚ަ͠ɼ|x′y| = |x| |y| ͳΒ ͹ͦΕΒ 2 ຊͷϕΫτϧ͸ฏߦͰ͋ΔɽϕΫτϧͷϊϧϜͱ಺ੵʹ͍ͭͯ͸ ҎԼͷؔ܎͕੒ཱ͢Δɽ 1
2. 1. |a|2 = (a · a) 2. |a + b|2 = |a|2 + |b|2 + 2 (a · b) 3. (αa · b) = (a · αb) = α (a · b) 4. |a|2 = 0 ⇐⇒ a = 0 ͜͜Ͱ α ͸εΧϥͰ͋Δɽ ϕΫτϧͷ಺ੵ͔Βίʔγʔɾγϡϫϧπ Cauchy–Schwarz ͷෆ౳ࣜ (a · b)2 ≤ |a|2 |b|2 (2) (∑ ai bi )2 ≤ (∑ a2 i ) (∑ b2 i ) (3) ͕ಋ͖ग़ͤΔɽ ূ໌͸ɼ೚ҙͷ࣮਺ t ʹରͯ͠ |a − tb|2 = |a|2 − 2t (a · b) + t2 |b|2 ≥ 0 (4) ͕ৗʹ੒ཱ͢ΔͨΊʹ͸ɼ൑ผࣜ D ͕ෛͰ͋Ε͹Α͍. ैͬͯ D = (a · b)2 − |a|2 |b|2 ≤ 0 (5) Ώ͑ʹ (a · b)2 ≤ |a|2 |b|2 (6) ͕੒Γཱͭɽ 1.3 ϕΫτϧؒͷڑ཭ 2 ຊͷϕΫτϧؒͷڑ཭Λ d(a, b) = |a − b| (7) ͱఆٛ͢Ε͹ɼ࣍ͷ͜ͱ͕੒Γཱͭɽ 1. d(a, b) ≥ 0 2. d(a, b) = 0 ⇐⇒ a = b 3. d(a, b) = d(b, a) 4. d(a, b) + d(b, c) ≥ d(a, c) ࠷ޙͷ΋ͷ͸ɼ͍ΘΏΔࡾ֯ෆ౳ࣜͱΑ͹ΕΔ΋ͷͰ͋Δɽ 2
3. 2 ߦྻ Matrix ਺஋Λॎԣͷۣܗʹ഑ஔͨ͠΋ͷΛߦྻͱ͍͏ɽA = (aij ) ͳͲͱද͞Ε Δɽ͜Ε͸ i ߦ j ྻ໨ͷཁૉ͕ aij Ͱ͋Δͱ͍͏ҙຯͰ͋Δɽߦ਺ͱྻ਺ͷ ౳͍͠ߦྻΛਖ਼ํߦྻͱ͍͏ɽ ͢΂ͯͷཁૉ͕ 0 Ͱ͋ΔߦྻΛྵߦྻͱ͍͍ 0 ͱଠจࣈΛ࢖ͬͯද͞ΕΔɽ ͋ΔߦྻͷߦͱྻͱΛೖΕ׵͑ͨߦྻΛసஔߦྻͱ͍͍ɼߦྻͷݞʹ t · ͨ͸ ′ Λ͚ͭͯද͢ A′ = (aji )ɽసஔߦྻʹ͸࣍ͷΑ͏ͳੑ࣭͕͋Δɽ • ( A′ )′ = A • (A + B)′ = A′ + B′ • (AB)′ = B′A′ సஔߦྻͱ΋ͱͷߦྻ͕౳͍͠ͱ͖ɼͦͷߦྻΛରশߦྻͱ͍͏ A′ = Aɽaii ཁૉ͕θϩͰͳ͘ɼଞͷཁૉ͕શͯθϩ aij = 0, (i ̸= j) Ͱ͋ΔߦྻΛର֯ߦ ྻͱ͍͏ɽD =       d1 0 · · · 0 0 d2 . . . ... 0 dn       ର֯ཁૉ͕શͯ 1 ͷର֯ߦྻΛ୯Ґ ߦྻͱ͍͍ I = (δij ) ͳͲͱද͢ɽ͜͜Ͱɼδ ͸ΫϩωοΧʔ kronecker ͷ σϧλͱݺ͹Ε δij = { 1 i = j ͷͱ͖ 0 i ̸= j ͷͱ͖ (8) Ͱ͋Δ. 2.1 ߦྻͷεΧϥഒ ߦྻͷεΧϥഒͱ͸ߦྻͷ֤੒෼ΛεΧϥഒͨ͠΋ͷͰ͋Γɼ࣍ͷΑ͏ͳ ԋࢉ͕ڐ͞ΕΔɽ͜͜Ͱ lambda ͸εΧϥͰ͋Δɽ • λA = Aλ • (λ + µ) A = λA + µA • λ (A + B) = λA + λB 2.2 ߦྻͷ࿨ ߦྻͷ࿨͸ରԠ͢Δ֤ཁૉΛՃ͑ͨ΋ͷͰ͋Δɽ͜ͷͨΊߦ਺ͱྻ਺͕౳ ͍͠ߦྻؒͰ͔͠ߦྻͷ࿨͸ఆٛ͞Εͳ͍ɽߦྻͷ࿨ʹ͸࣍ͷΑ͏ͳنଇ͕ ͋Δɽ 3
4. • A + B = B + A • (A + B) + C = A + (B + C) 2.3 ߦྻͷੵ 2 ͭͷߦྻͷੵ͸࣍ͷΑ͏ʹఆٛ͞ΕΔɽ Anm Bml =   m ∑ j=1 aij bjk   (9) Ұൠʹ AB ̸= BA Ͱ͋Δɽߦྻͷੵʹ͸࣍ͷΑ͏ͳੑ࣭͕͋Δɽ • (AB) C = A (BC) • A (B + C) = AB + AC • IA = AI = A 2.4 ߦྻࣜ Ұൠʹ d × d ͷਖ਼ํߦྻͷߦྻࣜ͸εΧϥͰ det (A) ͋Δ͍͸ |A| ͳͲͱ ද͢ɽ 2 ࣍ͷਖ਼ํߦྻ A = ( a1 b1 a2 b2 ) ͷߦྻࣜ͸ a = ( a1 a2 ) , b = ( b1 b2 ) ͱ͢Ε͹, a1 b2 − a2 b1 Ͱ͋Γ, ͜ͷ 2 ຊͷϕΫτϧʹΑͬͯ࡞ΒΕΔฏߦ࢛ ลܗͷ໘ੵʹ౳͍͠. |a| |b| sin θ = |a| |b| √ 1 − cos2 θ (10) = √ |a|2 |b|2 { |a|2 |b|2 − (a · b)2 |a|2 |b|2 } (11) = √ |a|2 |b|2 − (a · b)2 (12) = √ a2 1 b2 2 + a2 2 b2 1 − 2a1 b1 a2 b2 (13) = √ (a1 b2 − a2 b1 )2 (14) = a1 b2 − a2 b1 (15) (16) A ͷྻΛϕΫτϧͱߟ͑ɼͦΕΒͷϕΫτϧ͕ઢܗಠཱͰ͸ͳ͍ͱ͢Δͱ A ͷߦྻࣜ͸ 0 ʹͳΔɽٯߦྻ͕ଘࡏ͢ΔͨΊʹ͸ߦྻ͕ࣜ 0 Ͱ͋ͬͯ͸ ͳΒͳ͍ɽ Ұൠͷਖ਼ํߦྻͷߦྻࣜ͸ɼখߦྻల։ͱݺ͹ΕΔํ๏ͰܭࢉͰ͖ɼ࠶ؼ తʹఆٛͰ͖Δɽ͢ͳΘͪɼߦྻ A ͷ i ߦ j ྻΛআ͍ͯಘΒΕΔ m − 1 ߦ 4
5. m − i ྻͷখߦྻͷߦྻࣜʹ (−1)i+j Λ͔͚ͨ΋ͷΛ aij ͷ༨Ҽࢠ cofactor ͱ͍͍, Aij ͱදه͢Δɽ͜ͷͱ͖࣍ͷؔ܎͕੒Γཱͭɽ |A| = m ∑ j=1 aij Aij = m ∑ i=1 aij Aij (17) |Aij | ͕෼͔Ε͹ A ͷߦྻࣜ͸ୈ 1 ྻʹରͯ࣍͠ͷΑ͏ʹల։Ͱ͖Δɽ͜͜ Ͱූ߹͕ަޓʹมΘΔɽ |A| = a11 |A11 | − a22 |A21 | + a31 |A31 | − · · · ± md1 |Ad1 | (18) ͜ͷաఔ͸ΑΓখ͍͞ߦྻͷߦྻࣜʹରͯ͠࠶ؼతʹ܁Γฦ͠ద༻Ͱ͖Δɽ ߦྻࣜʹ͸࣍ͷΑ͏ͳੑ࣭͕͋Δɽ • ର֯ߦྻ, ࡾ֯ߦྻͷߦྻࣜ͸͢΂ͯͷର֯੒෼ͷੵʹ౳͍͠. |A| = n ∏ i=1 aii • A ͷ೚ҙͷ̎ྻʢߦʣΛೖΕ׵͑Δͱ |A| ͷූ߸͕มΘΔ • A′ = |A| • |AB| = |A| |B| • |AB| = |BA| • A−1 = |A|−1 2.5 ٯߦྻ ਖ਼ํߦྻͷߦྻ͕ࣜ 0 Ͱͳ͚Ε͹ٯߦྻ͕ଘࡏ͢Δɽߦྻ A ʹ͋Δߦྻ Λֻ͚ͯ౴͕୯ҐߦྻʹͳΔΑ͏ͳߦྻΛٯߦྻͱ͍͍ AA−1 = I ͱද͞ ΕΔɽٯߦྻʹ͸࣍ͷੑ࣭͕੒Γཱͭɽ • ( A−1 )−1 = A • ( A′ )−1 = ( A−1 )′ • (AB)−1 = B−1A−1 • (λA)−1 = 1 λ A−1 • ( A−1 + B−1 )−1 = A (A + B)−1 B = B (A + B)−1 A 5
6.   ࣍ͷٯߦྻΛٻΊΑ ( a b c d ) ( cos θ − sin θ sin θ cos θ )    1 1 1 0 1 1 0 0 1      ߦྻ M ͕ਖ਼ํߦྻͰͳ͍ͱ͖΍ɼߦྻ M ͷྻ͕ઢܗಠཱͰͳ͍ͨΊٯ ߦྻ M−1 ͕ଘࡏ͠ͳ͍ͱ͖࣍ͷٖࣅٯߦྻ M† ͕ఆٛͰ͖Δɽ M† = [ MtM ]−1 Mt (19) ٖࣅٯߦྻ͸ M†M = I ͕อূ͞ΕΔͷͰ࠷খ໰୊Λղ͘ͱ͖ศརͰ͋Δɽ 2.6 ௚ަߦྻ orthogonal matirx L′L = I, L′ = L−1 Λຬͨ͢ߦྻΛ௚ަߦྻͱ͍͏ɽ 2.7 τϨʔε Trace . d × d ͷਖ਼ํߦྻͷର֯ཁૉͷ࿨ΛτϨʔεͱ͍͍ tr (A) = n ∑ i=1 aii ͷΑ ͏ʹද͢ɽߦྻͷτϨʔεʹ͸࣍ͷੑ࣭͕͋Δɽ͜͜Ͱ c, d ͸εΧϥͰ͋Δɽ • tr (cA + dB) = c tr (A) + d tr (B) • tr (AB) = tr (BA) • tr ( A′A ) = m ∑ j=1 n ∑ i=1 a2 ij • tr A ( A′A )−1 A′ = m (ͨͩ͠ A ͸ rank A = m ͷ n × m ߦྻ) • tr (AB) = tr (BA) • tr (A′A) = 0 ⇐⇒ A = 0 • tr (A′B) ≤ √ tr (A′A) tr (B′B) Cauchy–Schwarz ͷෆ౳ࣜͷҰൠԽ 6
7. 3 ϕΫτϧۭؒ Vector space m ݸͷ 0 Ͱͳ͍ϕΫτϧ a1 , a2 , . . . , am Λ΋͍ͪͯ α1 a1 + α2 a2 + . . . + αm am = 0 ͕੒Γཱͭ৚݅Λߟ͑Δ. αi = 0 (i = 1, 2, . . . , m) ͷͱ͖͸໌Β͔ʹ੒Γཱ ͭ. ্͕ࣜ αi = 0 (i = 1, 2, . . . , m) Ҏ֎ͷࣗ໌Ͱͳ͍ղΛ΋ͭͱ͖, ϕΫτϧ a1 , a2 , . . . , am ͸ઢܗैଐʢ̍࣍ैଐʣlinearly dependent Ͱ͋Δͱ͍͏. ൓ ରʹ, ্͕ࣜ੒Γཱͭͷ͸ ai = 0 (i = 1, 2, . . . , m) ʹݶΔͱ͖, ઢܗಠཱʢ̍ ࣍ಠཱʣlinearly independent Ͱ͋Δͱ͍͏. a1 , a2 , . . . , am ͕ઢܗಠཱͳͱ͖, ai ̸= 0 Λ࢖ͬͯ bk = −ak /ai (k = 1, 2, . . . , m) Λ࡞ͬͯ ai = b1 a1 + · · · + bi−1 ai−1 + bi+1 ai+1 + · · · + bm am ͱͳΔ. ͕ͨͬͯ͠, ͋ΔϕΫτϧͷ૊͕ઢܗैଐͰ͋Δͱ͍͏͜ͱ͸, ͦͷ ૊ͷதͷ͋ΔϕΫτϧ͕ଞͷϕΫτϧͷઢܗ݁߹Ͱද͞ΕΔ͜ͱΛҙຯͯ͠ ͍Δ. 3.1 ઢܗ෦෼ۭؒ liner subspace m ݸͷઢܗಠཱͳϕΫτϧͷઢܗ݁߹ͷू߹Λ W = { b b = m ∑ i=1 ai ai } ͱදݱ͢Δ. m ࣍ݩϕΫτϧશମͷू߹Λ Em ͱ͢Δ, Em ͷ෦෼ू߹ W ͕ 1. a ∈ W, b ∈ W → a + b ∈ W 2. a ∈ W → ca ∈ W Λຬͨ͢ͱ͖, ͜ΕΛ m ࣍ݩͷઢܗ෦෼ۭؒͱ͍͏. 3.2 جఈ basis ೚ҙͷઢܗ෦෼ۭؒ W Ͱઢܗಠཱͳ m ݸͷϕΫτϧ͕ଘࡏ͠, (m + 1) ݸͷϕΫτϧ͸ઢܗैଐʹͳΔͱ͖ W ͷ࣍ݩ͸ r Ͱ͋Δͱ͍͍, dim W ͱ දه͢Δ. ্هͷ W ʹଐ͢Δઢܗಠཱͳ m ݸͷϕΫτϧΛۭؒ W ͷجఈ ͱ͍͏. ·ͨ, ۭؒ W ͸ m ݸͷϕΫτϧʹΑͬͯੜ੒͞ΕΔͱ͍͏. ͜ͷ ͜ͱΛ W = S (a1 , a2 , . . . , am ) = S (A) 7
8. ͱද͢. ॏཁͳఆཧ   m ࣍ݩͷ෦෼ۭؒ W ʹଐ͢Δ m ݸͷઢܗಠཱͳϕΫτϧΛ a1 , a2 , . . . , am ͱ͢Δͱ, W ʹଐ͢Δ೚ҙͷϕΫτϧ͸ a1 , a2 , . . . , am ͷઢܗ݁߹ͱͯͨͩ͠Ұ௨Γʹఆ·Δ.   ͭ·Γ, ઢܗ෦෼ۭؒͷ೚ҙͷϕΫτϧ͸, ͦͷۭؒΛఆΊΔجఈΛ༻͍ͯද ݱՄೳͰ͋Δ. Ұൠʹ, جఈͷఆΊํ͸Ұ௨ΓͰ͸ͳ͍. a1 , a2 , . . . , am ͕ W ͷجఈͰશ ͕ͯ௚ަ͢Δͱ͖, ௚ަجఈ orthogonal basis ͱ͍͏. ͞Βʹ, bj = aj / |aj | ͱ͢Δͱ, |bj | = 1 (1 ≤ j ≤ m) ͱͳΔ. ͜ΕΛਖ਼ن௚ަجఈ orthonormal basis ͱ͍͏. b1 , b2 , . . . .bm ͕ਖ਼ن௚ަجఈͷͱ͖ (bi · bj ) = δij Ͱ͋Δ. 3.3 ϥϯΫ rank A m × n ߦྻ A ͷߦϕΫτϧͷ͏ͪͰઢܗಠཱͳϕΫτϧͷݸ਺͸ A ͷྻϕΫτϧͷ͏ͪͰઢܗಠཱͳϕΫτϧͷݸ਺ʹ౳͍͠. ͜ͷઢܗಠཱ ͳϕΫτϧͷݸ਺ͷ͜ͱΛߦྻ A ͷϥϯΫͱ͍͏. • |A| = 0 ⇔ rank A < m (A ͸ m ࣍ͷਖ਼ํߦྻ) • rank A = rank A′ • rank (AB) ≤ max (rank A, rank B) • rank A = rank ( A′A ) = rank ( AA′ ) 3.4 ۭؒͷ෼ׂ 2 ͭͷϕΫτϧͷ૊ A = (a1 , a2 , · · · , ap ), B = (b1 , b2 , · · · , bq ) ɹʹΑͬ ͯੜ੒͞ΕΔ෦෼ۭؒΛͦΕͧΕ, VA = S (A), VB = S (B) ͱ͢Δ͜ͷͱ͖, ࿨ۭؒ: 2 ͭͷ෦෼ۭؒͷ࿨, VA + VB = {a + b |a ∈ VA , b ∈ VB } ͸ઢܗ ෦෼ۭؒʹͳΔ. ͜ΕΛ࿨ۭؒͱݺͼ, ࣍ͷΑ͏ʹදه͢Δ. VA∪B = VA + VB = S (A : B) 8
9. ੵۭؒ: 2 ͭͷ෦෼ۭؒͷڞ௨෦෼, VA∩B = {x |x = Aα = Bβ } ΋, ઢܗ෦෼ۭؒͰ͋Γ, ͜ΕΛੵۭؒͱݺͿ. ࣍ͷΑ͏ʹදه͢Δ. VA∩B = VA ∩ VB ௚࿨෼ղ: VA ∩ VB = {0} ͷͱ͖, ࿨ۭؒ VA∪B ͸ VA ͱ VB ͱʹ௚࿨෼ղ ͞Εͨͱݺͼ, ࣍ͷΑ͏ʹදه͢Δ. ·ͨ, VA ͱ VB ͸ಠཱͰ͋Δͱ΋ ͍͏. VA∪B = VA ⊕ VB ิۭؒ: શۭؒ En ͕ೋͭͷۭؒ V ͱ W ͱʹ௚࿨෼ղ͞ΕΔͱ͖, W ͸ V ͷิۭؒ complementary space ͱ͍͍ WC ͱදه͢Δ. ௚ަิۭؒ: W ʹଐ͢Δ೚ҙͷϕΫτϧ͕ V ʹଐ͢Δ೚ҙͷϕΫτϧͱ௚ ަ͢Δͱ͖, W ͸ V ͷ௚ަิۭؒ orthocoplementary space ͱ͍͍, W = V ⊥ ͱදه͢Δ. V ⊥ = {a |(a · b) = 0, ∃ b ∈ V } શۭؒ En ͕ r ݸͷಠཱͳۭؒ Wj (j = 1, 2, . . . , r) ʹ௚࿨෼ղ͞ΕΔ ͱ͖͸, ࣍ͷΑ͏ʹॻ͘. En = W1 ⊕ W2 ⊕ · · · ⊕ Wr ෦෼ۭؒͷ࣍ݩʹؔ͢Δఆཧ   1. dim (VA∪B ) = dimVA + dimVB − dimVA∩B 2. dim (VA ⊕ VB ) = dimVA + dimVB 3. dimV C = n − dimV   ·ͨ,   W = W1 ⊕ W2 ⊕ · · · ⊕ Wr ʹؚ·ΕΔ೚ҙͷϕΫτϧ x ͸ x = x1 + x2 + · · · + xr (xi ∈ Wi ) ͱͨͩҰ௨Γʹ෼ղ͞ΕΔ.   9
10. 3.5 ઢܗม׵ m ࣍ݩϕΫτϧ x Λ n ࣍ݩϕΫτϧ y ʹରԠͤ͞Δ͜ͱΛߟ͑, y = ϕ (x) ͱදه͢Δ. ࣍ͷੑ࣭Λ΋ͭม׵ ϕ Λઢܗม׵ͱ͍͏. 1. ϕ (αx) = αϕ (x). 2. ϕ (x + y) = ϕ (x) + ϕ (y). ೚ҙͷ m ࣍ݩϕΫτϧ x Λ n ࣍ݩϕΫτϧ y ʹରԠͤ͞Δઢܗม׵ ϕ ͸ m ݸͷ n ࣍ݩϕΫτϧʹΑͬͯߏ੒͞ΕΔ m×n ߦྻ A = (a1 , a2 , · · · , am ) Λ΋͍ͪͯ, y = Ax ͱදݱ͞ΕΔ. 3.6 ࣹӨߦྻ En = V ⊕ W ͷͱ͖, En ʹؚ·ΕΔ೚ҙͷϕΫτϧ x ͸ x = x1 + x2 (where, x1 ∈ V, x2 ∈ W) ͱͨͩҰ௨Γʹ෼ղ͞ΕΔ. ͜ͷͱ͖, x Λ x1 ʹҠ͢ม׵Λ W ʹԊͬͨ V ΁ͷࣹӨ projection ͱ͍͏. جຊੑ࣭   P ,Q ΛࣹӨߦྻͱ͢ΔͱҎԼͷੑ࣭͕੒Γཱͭ • ΂͖౳ੑ P 2 = P (ඞཁे෼৚݅) • ରশੑ x′ (P y) = x′P ′y = ( P x′y ) • ௚ަੑ P Q = 0 • ૬ิੑ P + Q = I   ఆཧ P ͕ࣹӨߦྻͳΒ͹ Q = I − P ΋ࣹӨߦྻͰ͋Δ. ূ໌: QQ = (I − P ) (I − P ) = I + P − P + P = I − P = Q ఆཧ P Q = QP = 0 ূ໌: P Q = P (I − P ) = P − P = 0 10
11. 4 ݻ༗஋ Eigen value ͨͱ͑͹, A = ( 1 1 2 1 2 1 ) ͱͯ͠, ( x y ) ΛͲ͜ʹ͔ࣸ͢Λߟ͑Δ. ( x′ y′ ) = A ( x y ) = ( x + y 2 x 2 + y ) ͜ͷߦྻ A ͸ y = x ͱ y = −x ͱ͍͏௚ઢͷํ޲ʹ͍ͭͯ͸֦େ, ॖখ͔͠ ͠ͳ͍. ͢ͳΘͪ A ( x x ) = 3 2 ( x x ) A ( x −x ) = 1 2 ( x −x ) ͦ͜Ͱ೚ҙͷϕΫτϧ x = (x, y)′ Λ͜ͷ 2 ͭͷϕΫτϧʹ෼ղͯ͠ߟ͑Δ. ͢ͳΘͪ, Ax = Ax1 + Ax2 = 3 2 x1 + 1 2 x2 1. A ʹ͸ 2 ͭͷൺྫ֦େʢॖখʣͷํ޲͕͋Δ 2. A ͸ͦͷ 2 ͭͷํ޲΁ͷ࡞༻ͷ࿨ͱͯ͠ද͞ΕΔ ͜ΕΛҰൠԽ͢Δͱ n ࣍ͷਖ਼ํߦྻ A ʹରͯ͠ Ax = λx (x ̸= 0) Λຬͨ͢ϕΫτϧ x Λ A ͷݻ༗ϕΫτϧ, λ Λ A ͷݻ༗஋ͱ͍͏. ͦ͜Ͱ, ҎԼͷ໰୊Λߟ͑Δ ໰୊ 1. ਖ਼ํߦྻ A ͷݻ༗ϕΫτϧͱݻ༗஋ΛٻΊΔํ๏Λߟ͑Δ ໰୊ 2. ٻΊͨݻ༗ϕΫτϧͰϕΫτϧۭؒͷجఈ͕࡞ΕΔΑ͏ͳߦྻΛ൑ ผ͢Δํ๏Λߟ͑Δ ఆཧ ਖ਼ํߦྻ A ͷݻ༗஋͸ λ ͷ୅਺ํఔࣜ det (A − λI) = 0 ͷࠜͰ͋ Δ. ٯʹ͜ͷํఔࣜͷࠜ͸ A ͷݻ༗஋Ͱ͋Δ ূ໌ x(̸= 0) Λ A ͷҰͭͷݻ༗ϕΫτϧ, λ0 ΛରԠ͢Δݻ༗஋ͱ͢Δͱ Ax = λ0 x Ax − λ0 x = 0 ͜Ε͸ x = (x1 , x2 , · · · , xn )′ ʹؔ͢Δ࿈ཱҰ࣍ํఔࣜͰ͋Δ. ͜Ε͕ 0 Ͱͳ ͍ղ x Λ΋͍ͬͯΔͷ͔ͩΒ, ͦͷ܎਺ͷߦྻࣜ͸ 0 Ͱͳ͚Ε͹ͳΒͳ͍. ٯʹ͋ΔεΧϥʔ λ0 ͕ det (A − λ0 I) = 0 Λຬͨͤ͹ (Ax − λ0 I) x = 0 ͸গͳ͘ͱ΋Ұͭͷ 0 Ͱͳ͍ղ x Λ࣋ͭ. ͦͷͱ͖ Ax = λ0 x (x ̸= 0) ͔ͩΒ, x ͸ A ͷݻ༗ϕΫτϧͰ͋Γ, λ0 ͸ରԠ͢Δݻ༗஋Ͱ͋Δ. 11
12. 4.1 ݻ༗஋ͷٻΊ͔ͨ det (A − λI) = 0 ͷͱ͖, ࣍ͷ֤ࠜ λ1 , λ2 ,· · · ,λn ʹରͯ͠࿈ཱํఔࣜ (A − λi I) = 0 Λͱ͍ͯ x1 ,x2 ,· · · ,xn ΛٻΊΕ͹Α͍. ྫ 1 A = ( 1 1 2 1 2 1 ) ͷݻ༗ํఔࣜ͸ det (A − λI) = 1 − λ 1 2 1 2 1 − λ = ( 1 − λ − 1 2 ) ( 1 − λ + 1 2 ) = 0 Ώ͑ʹ λ = 1 2 , 3 2 • λ1 = 1 2 ʹରԠ͢Δ A ͷݻ༗஋ϕΫτϧ͸ (A − λI) x1 = {( 1 1 2 1 2 1 ) − 1 2 ( 1 0 0 1 )} ( l m ) = ( 1 2 1 2 1 2 1 2 ) ( l m ) = ( 0 0 ) l + m = 0, ⇒ l = −m x = c ( 1 −1 ) (c ͸೚ҙͷ 0 Ͱͳ͍਺) • λ2 = 3 2 ʹରԠ͢Δ A ͷݻ༗஋ϕΫτϧ͸ (A − λI) x2 = {( 1 1 2 1 2 1 ) − 3 2 ( 1 0 0 1 )} ( l m ) = ( −1 2 1 2 1 2 −1 2 ) ( l m ) = ( 0 0 ) −l + m = 0, ⇒ l = m x = c ( 1 1 ) (c ͸೚ҙͷ 0 Ͱͳ͍਺) ྫ 2 ( 1 −1 1 1 ) ͷݻ༗஋ (ෳૉ਺ղ) ྫ 3 ( 3 1 −1 1 ) ͷݻ༗஋ (ॏࠜ) 12
13. 4.2 ݻ༗஋ͷ༻్ n ࣍ͷਖ਼ํߦྻ A ͷݻ༗஋Λର֯ཁૉʹ΋ͭߦྻΛ D ͱ͢Δ. ͜ͷ ݻ༗஋ʹରԠ͢Δݻ༗ϕΫτϧ (x1 ,x1 ,· · · ,xn ) Λฒ΂ͯͰ͖ͨߦྻΛ X = (x1 , x2 , · · · , xn ) ͱ͢Δͱ AX = XD ͱද͢͜ͱ͕Ͱ͖Δɽ·ͨಉ͜͡ͱ͕ͩ A ͸ X ͱ D ͱΛ༻͍ͯ A = XDX−1 ͱදݱͰ͖Δ. ͜ͷͱ͖ X ͷཁૉͰ͋ΔॎϕΫτϧ xi (i = 1, . . . , n), ͸શ ͯ௚ަ͢Δɽ͢ͳΘͪ೚ҙͷϕΫτϧ x ͕ x = x1 + x2 + · · · + xn ͷܗʹҰҙతʹ෼ղͰ͖Δ͜ͱΛҙຯ͢Δɽx → x1 , x → x2 , · · · , x → xn ʹ֤ʑࣹӨ͢Δૢ࡞ΛͦΕͧΕ P1 , P2 , · · · , Pn ͱॻ͘͜ͱʹ͢Δͱ x (= Ix) = P1 x + P2 x + · · · + Pn x ͕͢΂ͯͷ x ʹ͍ͭͯ੒ཱ͢Δɽࣸ૾͚ͩͷࣜʹͯ͠ΈΔͱ I = P1 + P2 + · · · + Pn ͕੒ཱ͢Δɽ͜ͷΑ͏ʹϕΫτϧʹ෼ղ͓ͯ͘͠ͱ Ax = Ax1 + Ax2 + · · · + An = λ1 x1 + λ2 x2 + · · · + λn xn ͱͳΔɽ͢ͳΘͪ A ͸ xi ํ޲ʹ͸ λi ഒ͢ΔҾ͖৳͹͠࡞༻Λ͍ͯ͠Δ͜ ͱʹͳΔɽ׵ݴ͢Ε͹ A ͸ݻ༗஋਺͚ͩͷҾ͖৳͹͠࡞༻ͷ߹੒ʹΑͬͯಘ ΒΕΔࣸ૾Ͱ͋ΔɽࣹӨԋࢉࢠ Pi Λ࢖ͬͯද͢ͱ Ax = λ1 P1 x + λ2 P2 x + · · · + λn Pn x ͱͳΔͷͰߦྻ͚ͩͷؔ܎ࣜʹ͢Δͱ A = λ1 P1 + λ2 P2 + · · · + λn Pn ͕੒ཱ͢Δ͜ͱʹͳΔɽI = P1 + P2 + · · · + Pn ͕੒ཱ͢ΔΑ͏ʹ Ax = λ1 P1 x + λ2 P2 x + · · · + λn Pn x ͷܗʹ෼ղ͢Δ͜ͱΛߦྻ A ͷࣹӨ෼ղ· ͨ͸εϖΫτϧ෼ղͱ͍͏ɽP1 , P2 , · · · , Pn ͸ͦΕͧΕ λ1 , λ2 , · · · , λn ʹର ͢Δݻ༗ۭؒ΁ͷࣹӨΛද͍ͯ͠Δɽ͜ͷ Pi Λ࢖͏ͱ A2 = λ2 1 P1 + λ2 2 P2 + · · · + λ2 n Pn 5A − I = (5λ1 + 1) P1 + (5λ2 + 1) P2 + · · · + (λn − 1) Pn ͳͲͷΑ͏ʹ A ͷଟ߲ࣜ΋ Pi Λ࢖ͬͯදݱͰ͖Δɽ ݻ༗஋ λi ʹର͢Δ (A − λi I) x = 0 Λຬͨ͢ϕΫτϧ x શମͷ͜ͱΛݻ ༗஋ λi ʹର͢Δݻ༗ۭؒͱ͍͏ɽ 13
14. 4.3 ݻ༗஋ͷੑ࣭ n ࣍ͷਖ਼ํߦྻ A ͷݻ༗஋ΛʢॏෳΛڐͯ͠ʣλ1 , λ2 , · · · , λn ͱ͢Δͱ 1. ݻ༗஋ͷੵ͸ߦྻࣜͰ͋Δ ∏ n i=1 λi = |A| 2. ݻ༗஋ͷ࿨͸ߦྻͷτϨʔεͰ͋Δ ∑ n i=1 λi = tr A 3. A′ ͷݻ༗஋͸ λ1 ,λ2 ,· · · ,λn 4. Ak (k = 1, 2, · · · ) ͷݻ༗஋͸ λk 1 ,λk 2 ,· · · ,λk n 5. f(n) Λ x ͷଟ߲ࣜͱ͢Δͱ͖ f(A) ͷݻ༗஋͸ f(λ1 ),f(λ2 ),· · · ,f(λn ) 5 ߦྻͷඍ෼ d ࣍ݩͷϕΫτϧɼ͢ͳΘͪ d ݸͷม਺ x1 , x2 , . . . , xd ΛͱΔؔ਺ f (x) Λ ߟ͑Δɽ͜ͷϕΫτϧʹؔͯ͠ f(·) ͷඍ෼͢ͳΘͪޯ഑͸ ∇f (x) = grad f (x) = ∂ (x) ∂x =       ∂f(x) ∂x1 ∂f(x) ∂x2 . . . ∂f(x) ∂xd       (20) Ͱܭࢉ͞ΕΔɽ ߦྻ M ͷཁૉ͕εΧϥ θ ͷؔ਺Ͱ͋Δͱ͖ɼߦྻ M Λ θ Ͱඍ෼͢Δͱ ࣍ͷߦྻΛಘΔɽ ∂M ∂θ =       ∂m11 ∂θ ∂m12 ∂θ · · · ∂m1d ∂θ ∂m21 ∂θ ∂m22 ∂θ · · · ∂m2d ∂θ . . . . . . ... . . . ∂mn1 ∂θ ∂mn2 ∂θ · · · ∂mnd ∂θ       (21) ͜ͷߦྻ M ͷٯߦྻ M−1 ͷඍ෼͸࣍ͷΑ͏ʹͳΔɽ ∂ ∂θ M−1 = −M−1 ∂M ∂θ M−1 (22) ߦྻ M ͓ΑͼϕΫτϧ y ͕ ϕΫτϧ x ʹಠཱͰ͋Δͱ͖ɼҎԼͷ౳ࣜ ͕੒Γཱͭɽ ∂ ∂x [Mx] = M (23) ∂ ∂x [ ytx ] = ∂ ∂x [ xty ] = y (24) 14
15. ∂ ∂x [ xtMx ] = [ M + Mt ] x (25) ͜͜ͰɼM ͕ରশߦྻͳΒ͹ ∂ ∂x [ xtMx ] = 2Mx (26) ͷΑ͏ʹ؆୯ʹͳΔɽ ∂tr ( XtY X ) ∂X = ( Y + Y t ) X (27) ∂ ln |X| ∂X = ( Xt )−1 (28) ∂ (xtMy) ∂M = xyt (29) ∂ ( xtMty ) ∂M = ytx (30) 6 ԋश໰୊ 1. ࣍ͷߦྻ͸ԿߦԿྻ͔ ( 1 3 0 7 −1 2 ) 2. ࣍ͷߦྻͷ 3 ߦ 2 ྻ໨ͷཁૉΛ౴͑Α            α β γ δ ϵ ζ η θ ι κ λ µ ν ξ o π ρ σ τ υ ϕ χ ψ ω            15
16. 3. A, B, C ͸ͦΕͧΕ 3 ߦ 2 ྻ,3 ߦ 3 ྻ,2 ߦ 3 ྻͷߦྻͱ͢ΔɽҎԼͷ ͏ͪߦྻͷੵ͕ఆٛͰ͖ͳ͍΋ͷʹ ×, Ͱ͖Δ΋ͷʹ͸ ⃝ Λ͚ͭΑɽ AA AB AC A′A A′B A′C BA BB BC B′A B′B B′C CA CB CC C′A C′B C′C AA′ AB′ AC′ A′A′ A′B′ A′C′ BA′ BB′ BC′ B′A′ B′B′ B′C′ CA′ CB′ CC′ C′A′ C′B′ C′C′ 4. ࣍ͷܭࢉΛͤΑ. (a) ( 7 5 1 −1 2 4 ) + ( 3 5 9 11 8 6 ) = (b) ( 2 0 5 )    −9 5 3    = (c) ( 1 2 3 1 ) ( x y ) = (d) ( x y ) ( 1 2 3 1 ) = (e)    ( 3 1 −2 1 )−1    ′ = 5. ࣍ͷࣜΛຬͨ͢ x ΛٻΊΑ.    2 0 7 0 1 0 1 2 1       −x −14x 7x 0 1 0 x 4x −2x    =    1 0 0 0 1 0 0 0 1    16
17. 6. A = ( 1 2 3 4 ) ͱͨ͠ͱ͖࣍ͷߦྻΛܭࢉͤΑ. (a) A3 = (b) A−1 = (c) A2 − 5A − 2I = 7. y =       0 5 8 9       , A =       1 1 1 2 1 3 1 4       ͱͨ͠ͱ͖࣍ͷߦྻΛܭࢉͤΑ. (a) A′A = (b) ( A′A )−1 = (c) θ = ( A′A )−1 A′y = (d) Q = A ( A′A )−1 A′ = (e) Q′ = (f) Q2 = (g) I − Q = 17
18. (h) (I − Q) (I − Q) = (i) y − Qy = y − Aθ = (j) y′ (I − Q) (I − Q) y = 8. y =       y1 y2 y3 y4       , A =       1 a1 1 a2 1 a3 1 a4       ͱͨ͠ͱ͖࣍ͷߦྻΛܭࢉͤΑ. (a) A′A = (b) ( A′A )−1 = (c) θ = ( A′A )−1 A′y = (d) Q = A ( A′A )−1 A′ = (e) Q′ = (f) Q2 = (g) I − Q = (h) (I − Q) (I − Q) = 18
19. (i) y − Qy = y − Aθ = (j) y′ (I − Q) (I − Q) y = 9. a = ( 3 2 ) , b = ( −3 1 ) , c = ( 3 −1 ) , d = ( −2 4 ) ͱ͢Δ. ͦΕͧΕͷ ϕΫτϧΛάϥϑʹॻ͚. 19
20. 10. ࿈ཱํఔࣜ { x + 2y = 6 3x − y = 4 ΛάϥϑΛ༻͍ͯղ͚. 11. ࣍ͷߦྻ A = ( 1 2 3 −1 ) ʹΑͬͯ ࣍ͷ֤఺ x = (x1 , x1 )′ ͸Ͳࣸ͜͞ ΕΔ͔άϥϑʹࣔͤ. (x,y) (x,y) (x,y) (0,0) (0,1) (0,2) (1,0) (1,1) (1,2) (2,0) (2,1) (2,2) 12. ࣍ͷߦྻ A = ( 1 2 3 −1 ) ʹΑͬͯ y = Ax ͸Ͳࣸ͜͞ΕΔ͔ϕΫτ ϧ y − x Λάϥϑʹࣔͤɽ͞Βʹ A ͷݻ༗஋ΛٻΊΑɽ 13. ࣍ͷߦྻ A = ( 2 0 0 2 ) ʹΑͬͯ y = Ax ͸Ͳࣸ͜͞ΕΔ͔ϕΫτϧ y − x Λάϥϑʹࣔͤɽ͞Βʹ A ͷݻ༗஋ΛٻΊΑɽ 14. ࣍ͷߦྻ A = ( −2 0 0 −2 ) ʹΑͬͯ y = Ax ͸Ͳࣸ͜͞ΕΔ͔ϕΫτ ϧ y − x Λάϥϑʹࣔͤɽ͞Βʹ A ͷݻ༗஋ΛٻΊΑɽ 15. ࣍ͷߦྻ A = ( 1 2 2 1 ) ʹΑͬͯ y = Ax ͸Ͳࣸ͜͞ΕΔ͔ϕΫτϧ y − x Λάϥϑʹࣔͤɽ͞Βʹ A ͷݻ༗஋ΛٻΊΑɽ 16. ࣍ͷߦྻ A = ( 0 1 −1 4 −5 4 ) ʹΑͬͯ y = Ax ͸Ͳࣸ͜͞ΕΔ͔ϕΫ τϧ y − x Λάϥϑʹࣔͤɽ͞Βʹ A ͷݻ༗஋ΛٻΊͤɽ 17. ࣍ͷߦྻ A = ( 2 1 1 2 ) ʹΑͬͯ y = Ax ͸Ͳࣸ͜͞ΕΔ͔ϕΫτϧ y − x Λάϥϑʹࣔͤɽ͞Βʹ A ͷݻ༗஋ΛٻΊͤɽ 18. ࣍ͷߦྻ A = ( 0 1 −26 −2 ) ʹΑͬͯ y = Ax ͸Ͳࣸ͜͞ΕΔ͔ϕΫ τϧ y − x Λάϥϑʹࣔͤɽ͞Βʹ A ͷݻ༗஋ΛٻΊͤɽ 19. ࣍ͷߦྻ A = ( 0 5 −2 2 ) ʹΑͬͯ y = Ax ͸Ͳࣸ͜͞ΕΔ͔ϕΫτ ϧ y − x Λάϥϑʹࣔͤɽ͞Βʹ A ͷݻ༗஋ΛٻΊͤɽ 20
21. 20. ࣍ͷߦྻ A = ( 0 1 −1 4 0 ) ʹΑͬͯ y = Ax ͸Ͳࣸ͜͞ΕΔ͔ϕΫτ ϧ y − x Λάϥϑʹࣔͤɽ͞Βʹ A ͷݻ༗஋ΛٻΊͤɽ 21. ࣍ͷߦྻ A = ( 2 1 1 1 2 ) ʹΑͬͯ y = Ax ͸Ͳࣸ͜͞ΕΔ͔ϕΫτϧ y − x Λάϥϑʹࣔͤɽ͞Βʹ A ͷݻ༗஋ΛٻΊͤɽ 21