Claudeに作らせた線形代数講義資料（数回分）

Claudeにシラバスを作成させ，さらに，それに基づき，第1回（ベクトルと幾何的直感），第2回（行列とその演算），第13回（固有値と固有ベクトル）の講義資料（beamer形式）を作成させた第１版です．

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1. ઢܗ୅਺ ୈ1ճ ϕΫτϧͱزԿత௚ײ ֶ෦ 1 ೥ 1 / 14
2. ຊ೔ͷ಺༰ 1 ϕΫτϧͱ͸ 2 ϕΫτϧͷԋࢉ 3 ϊϧϜͱڑ཭ 4 ಺ੵ 5 ·ͱΊͱԋश 2 / 14
3. ϕΫτϧͷ௚ײతΠϝʔδ ϕΫτϧ͸ʮେ͖͞ͱ޲͖Λ࣋ ͭྔʯ มҐɾ଎౓ɾྗ ˠ ෺ཧతͳྫ σʔλͷಛ௃ྔ ˠ ৘ใՊֶత ͳྫ ଟ߲ࣜɾؔ਺ ˠ ΑΓந৅తͳ ྫʢޙͷճͰʣ ͜ͷतۀͰ͸ɼ·ͣ Rnʢn ࣍ݩ࣮਺ ۭؒʣͷϕΫτϧΛѻ͏ɽ x y v 2 1.5 3 / 14
4. Rn ͷఆٛ Definition (Rn) n ݸͷ࣮਺Λॎʹฒ΂ͨ΋ͷΛ n ࣍ ݩྻϕΫτϧͱ͍͏ɿ x =      x1 x2 . . . xn      , xi ∈ R. n ࣍ݩϕΫτϧશମͷू߹Λ Rn ͱ ॻ͘ɽ ஫ҙ xi ͷॱং͸ҙຯΛ࣋ͭʢ਺ͷ૊ʣ ɽ ۩ମྫɿ R2 : ( 1 −2 ) R3 :   0 1 3   R4 :     1 0 0 1     4 / 14
5. ϕΫτϧͷՃ๏ Definition (Ճ๏) x, y ∈ Rn ʹରͯ͠ɼ x + y =    x1 + y1 . . . xn + yn    . ੒෼Ͱͷܭࢉྫ ( 1 3 ) + ( 2 −1 ) = ( 3 2 ) x y x y x + y 5 / 14
6. εΧϥʔഒ Definition (εΧϥʔഒ) x ∈ Rnɼc ∈ R ʹରͯ͠ɼ cx =    cx1 . . . cxn    . c > 1ɿಉํ޲ʹ৳ͼΔ 0 < c < 1ɿಉํ޲ʹॖΉ c < 0ɿٯํ޲ʹͳΔ c = 0ɿྵϕΫτϧ 0 x y x 2x −x 6 / 14
7. Ճ๏ɾεΧϥʔഒͷੑ࣭ x, y, z ∈ Rnɼc, d ∈ R ʹ͍ͭͯҎԼ͕੒Γཱͭɿ ԋࢉͷੑ࣭ʢ8 ͭʣ 1 x + y = y + x 2 (x + y) + z = x + (y + z) 3 x + 0 = x 4 x + (−x) = 0 5 c(x + y) = cx + cy 6 (c + d)x = cx + dx 7 (cd)x = c(dx) 8 1 · x = x ͜ͷ 8 ͭͷੑ࣭͕ʮϕΫτϧۭؒʯͷެཧʹͳΔʢୈ 10 ճʣ ɽ 7 / 14
8. ϊϧϜʢϕΫτϧͷେ͖͞ʣ Definition (ϊϧϜ) x ∈ Rn ͷϊϧϜʢ௕͞ʣΛ ∥x∥ = √ x2 1 + x2 2 + · · · + x2 n ͱఆٛ͢Δɽ Example x = ( 3 4 ) ͷͱ͖ ∥x∥ = √ 9 + 16 = 5ɽ ୯ҐϕΫτϧ ∥x∥ = 1 ͷͱ͖ɼx Λ୯ҐϕΫτϧͱ͍͏ɽ x ̸= 0 ͷͱ͖ɼ x ∥x∥ ͸ x ͱಉํ޲ͷ୯ҐϕΫτϧʢਖ਼نԽʣ ɽ 8 / 14
9. ಺ੵͷఆٛ Definition (಺ੵ) x, y ∈ Rn ͷ಺ੵʢυοτੵʣΛ x · y = ⟨x, y⟩ = x1 y1 + x2 y2 + · · · + xn yn ͱఆٛ͢Δɽ Example   1 2 3   ·   4 0 −1   = 1 · 4 + 2 · 0 + 3 · (−1) = 1. ಺ੵͱϊϧϜͷؔ܎ ∥x∥2 = x · x = x2 1 + · · · + x2 n . 9 / 14
10. ಺ੵͷزԿతҙຯ Theorem (಺ੵͱ֯౓) x, y ∈ Rnʢx, y ̸= 0ʣ ɼ྆ऀͷͳ֯͢Λ θ ∈ [0, π] ͱ͢Δͱɼ x · y = ∥x∥ ∥y∥ cos θ. ॏཁͳಛผͳ৔߹ɿ θ = 0 ⇒ x · y = ∥x∥ ∥y∥ ʢಉ ํ޲ʣ θ = π/2 ⇒ x · y = 0 ʢ௚ަʣ θ = π ⇒ x · y = − ∥x∥ ∥y∥ ʢٯํ޲ʣ x y θ 10 / 14
11. ίʔγʔɾγϡϫϧπෆ౳ࣜ Theorem (ίʔγʔɾγϡϫϧπͷෆ౳ࣜ) ೚ҙͷ x, y ∈ Rn ʹରͯ͠ |x · y| ≤ ∥x∥ ∥y∥ . ౳߸੒ཱ ⇔ x ͱ y ͕ઢܗैଐʢҰํ͕ଞํͷεΧϥʔഒʣ ɽ ূ໌ͷํ਑. cos θ ͷఆ͔ٛΒ | cos θ| ≤ 1 Λ༻͍Δɽ ʢୈ 13 ճҎ߱ɼݻ༗஋Λ༻ ͍ͨผূ໌΋Մೳɽ ʣ Ԡ༻ɿࡾ֯ෆ౳ࣜ ∥x + y∥ ≤ ∥x∥ + ∥y∥ . ʢίʔγʔɾγϡϫϧπ͔Βಋ͚Δɽ ʣ 11 / 14
12. ௚ަͱਖ਼ن௚ަ Definition x · y = 0 ͷͱ͖ɼx ͱ y ͸௚ަ͢Δͱ͍͏ʢx ⊥ yʣ ɽ ∥x∥ = 1 ͔ͭ x ⊥ yɼ∥y∥ = 1 ͷͱ͖ɼਖ਼ن௚ަͱ͍͏ɽ Example (R2 ͷඪ४جఈ) e1 = ( 1 0 ) , e2 = ( 0 1 ) ͸ਖ਼ن௚ަɿe1 · e2 = 0ɼ∥e1 ∥ = ∥e2 ∥ = 1ɽ ਖ਼ن௚ަܥ͸ୈ 14 ճʮର֯ԽʯͰॏཁͳ໾ׂΛՌͨ͢ɽ 12 / 14
13. ຊ೔ͷ·ͱΊ ॏཁࣄ߲ 1 Rn ͷϕΫτϧɿn ݸͷ࣮਺ͷ૊ 2 Ճ๏ɾεΧϥʔഒɿ੒෼͝ͱͷԋࢉɼ8 ͭͷੑ࣭ 3 ϊϧϜɿ∥x∥ = √∑ x2 i ɼਖ਼نԽ 4 ಺ੵɿx · y = ∑ xi yi = ∥x∥ ∥y∥ cos θ 5 ௚ަɿx · y = 0 ࣍ճʢୈ 2 ճʣ ߦྻͱͦͷԋࢉʢՃ๏ɾεΧϥʔഒɾੵɾసஔʣ 13 / 14
14. ԋश໰୊ 1 x =   2 −1 3  ɼy =   0 4 1   ʹ͍ͭͯܭࢉͤΑɽ (a) 2x − 3y (b) ∥x∥ɼ∥y∥ (c) x · y 2 ࣍ͷϕΫτϧΛਖ਼نԽͤΑɿa =   1 1 1  ɽ 3 x = ( 1 2 ) ɼy = ( a −1 ) ͕௚ަ͢Δͱ͖ɼa ΛٻΊΑɽ 4 R3 ʹ͓͍ͯɼe1 =   1 0 0  ɼe2 =   0 1 0  ɼe3 =   0 0 1   ͕ਖ਼ن௚ ަܥͰ͋Δ͜ͱΛ֬ೝͤΑɽ 14 / 14
15. ઢܗ୅਺ ୈ2ճ ߦྻͱͦͷԋࢉ ֶ෦ 1 ೥ 1 / 11
16. ຊ೔ͷ಺༰ 1 ߦྻͷఆٛ 2 ߦྻͷԋࢉɿՃ๏ɾεΧϥʔഒ 3 ߦྻͷੵ 4 సஔߦྻͱରশߦྻ 5 ·ͱΊͱԋश 2 / 11
17. ߦྻͱ͸ Definition (m × n ߦྻ) m × n ݸͷ࣮਺Λ m ߦ n ྻʹฒ΂ͨ΋ͷΛ m × n ߦྻͱ͍͏ɿ A =      a11 a12 · · · a1n a21 a22 · · · a2n . . . . . . ... . . . am1 am2 · · · amn      = (aij ). aij ɿୈ i ߦɾୈ j ྻͷ੒෼ɽ Example 1 2 3 4 5 6 ͸ 2 × 3 ߦྻɽa12 = 2ɼa23 = 6ɽ 3 / 11
18. ಛघͳߦྻ ਖ਼ํߦྻʢm = nʣ 1 2 3 4 (2 × 2) ྵߦྻ O 0 0 0 0 ୯Ґߦྻ In I3 =   1 0 0 0 1 0 0 0 1   ର֯ߦྻ   d1 0 0 0 d2 0 0 0 d3   ্ࡾ֯ߦྻ   ∗ ∗ ∗ 0 ∗ ∗ 0 0 ∗   Լࡾ֯ߦྻ   ∗ 0 0 ∗ 0 ∗ ∗   4 / 11
19. ߦྻͷՃ๏ɾεΧϥʔഒ Definition ಉ͡αΠζͷߦྻ A = (aij )ɼB = (bij ) ͱ c ∈ R ʹରͯ͠ɿ Ճ๏ɿA + B = (aij + bij )ʢ੒෼͝ͱͷ࿨ʣ εΧϥʔഒɿcA = (caij )ʢ֤੒෼ʹ c Λ͔͚Δʣ Example 1 2 3 4 + 0 −1 2 1 = 1 1 5 5 , 3 1 2 3 4 = 3 6 9 12 ஫ҙ αΠζ͕ҧ͏ߦྻಉ࢜ͷՃ๏͸ఆٛ͞Εͳ͍ɽ 5 / 11
20. ߦྻͷੵʢఆٛʣ Definition (ߦྻͷੵ) Aɿm × k ߦྻɼBɿk × n ߦྻͷͱ͖ɼੵ C = AB ͸ m × n ߦྻ Ͱɼ(i, j) ੒෼͸ cij = k ℓ=1 aiℓ bℓj = ai1 b1j + ai2 b2j + · · · + aik bkj . A m×k B k×n = C m×n ಺ଆͷαΠζ͕Ұக͠ͳ͍ͱੵ ͸ఆٛ͞Εͳ͍ɽ A m × k B k × n = C m × n 6 / 11
21. ߦྻͷੵʢܭࢉྫʣ A = 1 2 3 4 , B = 0 1 2 −1 AB = 1 · 0 + 2 · 2 1 · 1 + 2 · (−1) 3 · 0 + 4 · 2 3 · 1 + 4 · (−1) = 4 −1 8 −1 AB ̸= BA ͕Ұൠʹ੒Γཱͭʂ BA = 0 · 1 + 1 · 3 0 · 2 + 1 · 4 2 · 1 + (−1) · 3 2 · 2 + (−1) · 4 = 3 4 −1 0 ߦྻͷੵ͸ඇՄ׵ɿAB ̸= BAʢҰൠʹʣ ɽ 7 / 11
22. ߦྻͷੵͷੑ࣭ A, B, C Λద੾ͳαΠζͷߦྻɼc ∈ R ͱ͢Δɽ ੒Γཱͭੑ࣭ (AB)C = A(BC) ʢ݁߹ଇʣ A(B + C) = AB + ACɼ(A + B)C = AC + BC ʢ෼഑ଇʣ c(AB) = (cA)B = A(cB) Im A = A = AIn ʢAɿm × nʣ ʢ୯Ґߦྻʣ OA = OɼAO = O ʢྵߦྻʣ ੒Γཱͨͳ͍ੑ࣭ AB = BAʢҰൠʹෆ੒ཱɿඇՄ׵ʣ AB = O ⇒ A = O ·ͨ͸ B = Oʢෆ੒ཱʣ AC = BC, C ̸= O ⇒ A = Bʢෆ੒ཱʣ 8 / 11
23. సஔߦྻ Definition (సஔߦྻ) A = (aij )ʢm × nʣͷసஔߦྻ ATʢn × mʣΛ (AT)ij = aji ͰఆΊΔʢߦͱྻΛೖΕସ͑Δʣ ɽ Example A = 1 2 3 4 5 6 ⇒ AT =   1 4 2 5 3 6   సஔͷੑ࣭ (AT)T = A (A + B)T = AT + BT (cA)T = cAT (AB)T = BTATʢॱং͕ٯʣ Definition (ରশߦྻɾަ୅ߦྻ) 9 / 11
24. ຊ೔ͷ·ͱΊ ॏཁࣄ߲ 1 ߦྻͷ (i, j) ੒෼ɿaij ʢߦ͕ઌɼྻ͕ޙʣ 2 Ճ๏ɾεΧϥʔഒɿ੒෼͝ͱʢαΠζҰக͕ඞཁʣ 3 ੵ ABɿ(i, j) ੒෼ = A ͷୈ i ߦͱ B ͷୈ j ྻͷ಺ੵ 4 AB ̸= BAʢҰൠʹʣ ɼྵҼࢠ͕ଘࡏ͢Δ 5 సஔɿ(AB)T = BTATʢॱংٯసʣ ࣍ճʢୈ 3 ճʣ ߦྻͷੵͷੑ࣭ͷਂ۷Γͱಛघߦྻʢ୯Ґߦྻɾର֯ߦྻʣͷੑ࣭ 10 / 11
25. ԋश໰୊ 1 A = 2 −1 0 3 ɼB = 1 2 −1 0 ʹ͍ͭͯɿ (a) ABɼBA Λܭࢉ͠ɼAB ̸= BA Λ֬ೝͤΑɽ (b) A2 = AA ΛܭࢉͤΑɽ 2 A = 1 2 3 4 ʹ͍ͭͯɼAT ΛٻΊɼAAT ͕ରশߦྻͰ͋Δ͜ ͱΛ֬ೝͤΑɽ 3 AB = Oʢྵߦྻʣ͕ͩ A ̸= OɼB ̸= O ͱͳΔ 2 × 2 ߦྻ Aɼ B ͷྫΛҰͭ࡞Εɽ 4 n × n ߦྻ A ʹ͍ͭͯɼA + AT ͸ରশߦྻɼA − AT ͸ަ୅ ߦྻͰ͋Δ͜ͱΛࣔͤɽ 11 / 11
26. ઢܗ୅਺ ୈ13ճ ݻ༗஋ͱݻ༗ϕΫτϧ ֶ෦ 1 ೥ 1 / 15
27. ຊ೔ͷ಺༰ 1 ݻ༗஋ɾݻ༗ϕΫτϧͱ͸ 2 ಛੑํఔࣜ 3 ݻ༗஋ɾݻ༗ϕΫτϧͷܭࢉ 4 ݻ༗ۭؒͱॏෳ౓ 5 ੑ࣭ͱԠ༻΁ͷ઀ଓ 6 ·ͱΊͱԋश 2 / 15
28. ߦྻ͕ʮ৳ॖ͚ͩ͢Δʯํ޲ ߦྻ A Λֻ͚ΔͱɼҰൠʹํ޲͕ มΘΔɿ Ax ̸= cx ʢํ޲͕มΘΔʣ ͔͠͠ɼಛผͳϕΫτϧ v ̸= 0 ʹ ରͯ͠ Av = λv ͕੒Γཱͭ͜ͱ͕͋Δɽ → v ͷํ޲͸มΘΒͣɼλ ഒʹ৳ ॖ͢Δ͚ͩɽ x y v Av = 2v x Ax ʢํ޲มΘΔʣ 3 / 15
29. ݻ༗஋ɾݻ༗ϕΫτϧͷఆٛ Definition (ݻ༗஋ɾݻ༗ϕΫτϧ) n × n ߦྻ A ʹର͠ɼ Av = λv, v ̸= 0 Λຬͨ͢εΧϥʔ λ ∈ Rʢ·ͨ͸ CʣΛ ݻ༗஋ɼv Λ λ ʹରԠ͢ Δ ݻ༗ϕΫτϧͱ͍͏ɽ ௚ײతͳҙຯ ݻ༗ϕΫτϧɿߦྻ A ͕ʮํ޲Λม͑ͳ͍ʯϕΫτϧ ݻ༗஋ɿͦͷͱ͖ͷεέʔϧഒ཰ʢλ < 0 ͳΒٯ޲͖ʣ ஫ҙ v = 0 ͸ఆ͔ٛΒআ͘ʢࣗ໌͗͢ΔͨΊʣ λ = 0 ͸ݻ༗஋ʹͳΓಘΔʢAv = 0ɼA ͸ਖ਼ଇͰͳ͍ʣ 4 / 15
30. ಛੑํఔࣜͷಋग़ Av = λv Λมܗ͢Δɿ Av = λv ⇐⇒ Av − λv = 0 ⇐⇒ (A − λI)v = 0 v ̸= 0 ͱͳΔղ͕ଘࡏ͢ΔͨΊͷ৚݅ɿ (A − λI)v = 0 ͕ v ̸= 0 ͷղΛ࣋ͭ ⇐⇒ det(A − λI) = 0. Definition (ಛੑํఔࣜɾಛੑଟ߲ࣜ) p(λ) = det(A − λI) = 0 Λ ಛੑํఔࣜɼp(λ) Λ ಛੑଟ߲ࣜͱ͍͏ɽp(λ) ͸ λ ͷ n ࣍ଟ ߲ࣜɽ 5 / 15
31. 2 × 2 ͷ৔߹ͷެࣜ A = ( a b c d ) ͷͱ͖ɿ det(A − λI) = det ( a − λ b c d − λ ) = (a − λ)(d − λ) − bc. ల։͢Δͱɿ p(λ) = λ2 − (a + d)λ + (ad − bc) = λ2 − tr(A)λ + det(A). 2 × 2 ͷಛੑํఔࣜ λ2 − tr(A) λ + det(A) = 0 tr(A) = a + dɿτϨʔεʢର֯੒෼ͷ࿨ʣ 6 / 15
32. ܭࢉखॱ ݻ༗஋ɾݻ༗ϕΫτϧΛٻΊΔखॱ 1 ಛੑํఔࣜΛղ͘ɿdet(A − λI) = 0 ͷࠜ λ1 , λ2 , . . . ΛٻΊΔɽ 2 ݻ༗ϕΫτϧΛٻΊΔɿ֤ݻ༗஋ λk ʹରͯ͠ɼ (A − λk I)v = 0 Λղ͘ʢ૟͖ग़͠๏ʣ ɽ 3 ݻ༗ۭؒɿker(A − λk I) = {v | (A − λk I)v = 0}ɽ 7 / 15
33. ܭࢉྫʢ2 × 2ʣ A = ( 3 1 1 3 ) ͷݻ༗஋ɾݻ༗ϕΫτϧΛٻΊΔɽ Step 1ɿಛੑํఔࣜ det(A−λI) = det ( 3 − λ 1 1 3 − λ ) = (3−λ)2 −1 = λ2 −6λ+8 = 0. ∴ λ1 = 2, λ2 = 4. Step 2ɿλ1 = 2 ͷݻ༗ϕΫτϧ (A − 2I)v = ( 1 1 1 1 ) v = 0 =⇒ v = t ( 1 −1 ) (t ̸= 0). 8 / 15
34. ܭࢉྫʢଓ͖ʣ Step 2’ɿλ2 = 4 ͷݻ༗ϕΫτϧ (A − 4I)v = ( −1 1 1 −1 ) v = 0 =⇒ v = t ( 1 1 ) (t ̸= 0). ݁Ռͷ·ͱΊ ݻ༗஋ ݻ༗ϕΫτϧʢͷجఈʣ زԿతҙຯ λ1 = 2 ( 1 −1 ) ࣼΊࠨԼํ޲Λ 2 ഒ λ2 = 4 ( 1 1 ) ࣼΊӈ্ํ޲Λ 4 ഒ x y λ2 λ1 9 / 15
35. ݻ༗ۭؒ Definition (ݻ༗ۭؒ) ݻ༗஋ λ ʹର͢Δ ݻ༗ۭؒΛ Eλ = ker(A − λI) = {v ∈ Rn | Av = λv} ͱఆٛ͢Δɽ ʢ0 ΛؚΉ Rn ͷ෦෼ۭؒɽ ʣ ॏෳ౓ ୅਺తॏෳ౓ɿಛੑଟ߲ࣜʹ͓͚Δࠜ λ ͷॏෳ਺ زԿతॏෳ౓ɿdim Eλ ৗʹɿزԿతॏෳ౓ ≤ ୅਺తॏෳ౓ Example (3 × 3 ͷྫ) ಛੑํఔࣜ (λ − 1)2(λ − 3) = 0ɿλ = 1 ͷ୅਺తॏෳ౓͸ 2ɼ λ = 3 ͸ 1ɽ 10 / 15
36. ॏෳݻ༗஋ͷྫ A = ( 2 1 0 2 ) ͷݻ༗஋ɾݻ༗ϕΫτϧΛٻΊΔɽ ಛੑํఔࣜɿ det(A − λI) = (2 − λ)2 = 0 =⇒ λ = 2 ʢ୅਺తॏෳ౓ 2ʣ ݻ༗ۭؒɿ (A − 2I)v = ( 0 1 0 0 ) v = 0 =⇒ v = t ( 1 0 ) ஫ҙɿର֯ԽෆՄೳͳ৔߹ dim E2 = 1ʢزԿతॏෳ౓ 1ʣ< ୅਺తॏෳ౓ 2ɽ ͜ͷߦྻ͸ର֯ԽͰ͖ͳ͍ʢୈ 14 ճͰৄ͘͠ѻ͏ʣ ɽ 11 / 15
37. ݻ༗஋ͷجຊతͳੑ࣭ ੑ࣭ A Λ n × n ߦྻɼλ1 , . . . , λn Λͦͷݻ༗஋ʢॏෳࠐΈʣͱ͢Δͱɿ 1 tr(A) = λ1 + λ2 + · · · + λn 2 det(A) = λ1 · λ2 · · · λn 3 A ͕ਖ਼ଇ ⇐⇒ ͢΂ͯͷݻ༗஋ ̸= 0 4 ҟͳΔݻ༗஋ʹରԠ͢Δݻ༗ϕΫτϧ͸ઢܗಠཱ Example (ݕࢉʹ࢖͑Δ) ઌ΄Ͳͷྫ A = ( 3 1 1 3 ) ɼλ1 = 2, λ2 = 4ɿ tr(A) = 6 = 2 + 4 ✓, det(A) = 8 = 2 × 4 ✓. 12 / 15
38. ର֯Խ΁ͷ઀ଓʢϓϨϏϡʔʣ ҟͳΔݻ༗஋ λ1 , . . . , λn Λ࣋ͭߦྻ A ͸ର֯ԽՄೳɿ P−1AP =    λ1 ... λn    = Λ ͜͜Ͱ P ͸ݻ༗ϕΫτϧΛྻʹฒ΂ͨߦྻɽ ର֯Խͷҙຯʢୈ 14 ճͰৄ͘͠ʣ Ak = PΛkP−1ʢߦྻͷ k ৐͕؆୯ʹܭࢉͰ͖Δʣ Ϛϧίϑ࿈࠯ͷ௕ظڍಈͷղੳ ࣮ରশߦྻ → ࣮ݻ༗஋ͷΈɾ௚ަର֯ԽՄೳʢୈ 14 ճʣ 13 / 15
39. ຊ೔ͷ·ͱΊ ॏཁࣄ߲ 1 Av = λvʢv ̸= 0ʣ ɿλ ͕ݻ༗஋ɼv ͕ݻ༗ϕΫτϧ 2 ܭࢉखॱɿdet(A − λI) = 0 → ݻ༗஋ → (A − λI)v = 0 → ݻ ༗ϕΫτϧ 3 ݻ༗ۭؒ Eλ = ker(A − λI) 4 ݕࢉɿtr(A) = ∑ λi ɼdet(A) = ∏ λi 5 ॏෳݻ༗஋ɿ୅਺తॏෳ౓ ≥ زԿతॏෳ౓ ࣍ճʢୈ 14 ճʣ ର֯ԽɿP−1AP = Λ ͷ৚݅ͱܭࢉɼ࣮ରশߦྻͷ௚ަର֯Խ 14 / 15
40. ԋश໰୊ 1 ࣍ͷߦྻͷݻ༗஋ͱݻ༗ϕΫτϧΛٻΊΑɿ (a) ( 4 1 2 3 ) (b) ( 1 −2 −2 1 ) 2 A = ( 0 −1 1 0 ) ͷಛੑํఔࣜΛٻΊɼ࣮਺ͷݻ༗஋͕ଘࡏ͠ͳ ͍͜ͱΛ֬ೝͤΑɽ ʢ͜ͷߦྻͷزԿతҙຯ͸Կ͔ʁʣ 3 λ ͕ A ͷݻ༗஋ɼv ͕ରԠ͢Δݻ༗ϕΫτϧͷͱ͖ɼλ2 ͸ A2 ͷݻ༗஋Ͱ͋Δ͜ͱΛࣔͤɽ 4 ݻ༗஋ͱݻ༗ϕΫτϧͷੑ࣭͔Βɼtr(A) = 6ɼdet(A) = 8 Ͱ ͋Δ 2 × 2 ߦྻͷݻ༗஋ΛٻΊΑɽ 15 / 15