矩阵理论与应用(张跃辉)习题参考解答 (上海交大)
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इᆔູࠧ E12 ) ֻ֥ 2 ྛაֻ j ྛđᆃਆྛ൞ 0 ྛđܣ෮֤इᆔಯ൞ E12 . ၹՎ Eij (i = j ) ა E12 ཌྷර. ᄜᆣ aE12 (a = 0 ა E12 ཌྷර. ູՎđ౼ P = I + (a − 1)E22 ࠧॖ. োරॖᆣđֻ၂ᇕԚ֩इᆔ္дՎཌྷර. 19. ഡइᆔ A ડቀٚӱ A2 − A + 2I = 0, ໙ A ॖၛؓ࢘߄ગ? ູહ? ࡼЧี၂Ϯ߄. ࢳğ(A − I/2)2 + 7I/4 = 0, ࠧ (A − I/2 + bI )(A − I/2 − bI ) = 0(ఃᇏ b2 = −7/4)đၹ Վ A ൞ॖؓ࢘߄इᆔ. ၂Ϯֹđ (A − xI )(A − yI ) = 0, ᄵ A ॖၛؓ࢘߄. 20. ᆣૼ:(1) Hermite इᆔ֥หᆘᆴनູൌඔ, උႿ҂หᆘᆴ֥หᆘཟਈдՎᆞࢌ. (2) Hermite इᆔ A ൞ᆞקइᆔ ⇐⇒ թᄝॖ༯࢘इᆔ L ֤ A = LL∗ . ᆣૼğ(1) ഡ A∗ = A Ⴕ၂۱หᆘᆴ a ∈ C, ཌྷႋ֥หᆘཟਈູ α. ᄵ Aα = aα, ¯α∗ . ෮ၛ α∗ A∗ α = a ¯α∗ α. ቐ =α∗ Aα = a(α∗ α), طα∗ α = 0, ܣαT AT = aαT , ࣉ طα∗ A∗ = a ܣa=a ¯. ࠧ a ൞ൌඔ. ᄜഡ A∗ = A ߎႵ၂۱หᆘᆴ b = a ∈ C, ཌྷႋႿ b ֥หᆘཟਈູ β , ࠧ Aβ = bβ . ၹ ູ aα∗ β = (Aα)∗ β = α∗ A∗ β = α∗ (A∗ β ) = α∗ (Aβ ) = bα∗ β , ၹՎ (a − b)α∗ β = 0, ႮႿ a − b = 0, ܣα∗ β = 0đࠧ α ა β ᆞࢌ. 21. ഡ V = { ෮Ⴕᆞൌඔ }, F = R ൞ൌඔთ. קၬ V ᇏ֥ࡆمᄎෘູ x ⊕ y = xy (ࠧ๙ ӈ֥ൌඔӰק ;)مၬ V ᇏჭა F ᇏඔ֥ඔӰᄎෘູ k • x = xk (๙ӈ֥ૢᄎෘ). ᄵ (V, ⊕, •) ൞ൌཌྟॢࡗ. ᆣૼğᆰࢤဒᆣࠧॖ. 22. ഡ V = C \ {−1}. ০Ⴈ௴๙ࡆ௴ބم๙Ӱקمၬ V ഈ֥ࡆمo pೂ༯: a b = a + b + ab. ᆣૼ ડቀཌྟॢࡗ֥ࡆ֥مಆ่҆ࡱ. ࣉ၂҄, ܒᄯگඔა V ᇏཟਈ֥၂۱oඔӰp♥, ֤ (V, , ♥) ൞ R ഈ֥ཌྟॢࡗ. ౨۳ԛھཌྟॢࡗ֥၂ቆࠎ. ࢳğ൮༵ࡆੰߐࢌ֥مཁಖડቀčၹູ௴๙ࡆمაӰمनऎႵࢌߐੰĎĠࢲੰކဒᆣೂ ༯ğ (a b) c = (a + b + ab) c = a + b + ab + c + (a + b + ab)c = a + b + c + ab + ac + bc + abc ॖՎᄎෘაՑܱ. ਬჭູ 0Ġ0 a = 0 + a + 0a = a. ഡ a = −1 ֥ڵჭູ x, ᄵ 0 = a x = a + x + ax, ෮ၛ x = −a/(1 + a)(ႮՎॖᆩԢಀ −1 ֥сေྟ). Ⴎഈॖᆩ ಒູࡆم.
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√ (2) ( 2)n
;(ᇿğՎॖႮ (1) ֤
2. ᆣૼ: აၩ n ࢨٚᆔॖࢌߐ֥इᆔс൞Ղਈइᆔ λI . ᆣૼ: ഡ A = (aij ) აၩइᆔॖߐđᄵ A აࠎЧइᆔ Eij ॖߐ, ࠧ AEij = Eij A. ᇿ ၩ AEij ൞ֻ j ਙູ A ֻ֥ i ਙఃჅਙनູ 0 ֥इᆔđEij A ൞ֻ i ྛູ A ֻ֥ j ྛఃჅྛ नູ 0 ֥इᆔđႮՎॖᆩ A ֥٤ؓ࢘ჭनູ 0đ࢘ؓطჭनડቀ aii = ajj đࠧ A ൞Ղਈ इᆔ. 3. ০ႨԚ֩эߐ A−1 B ࠣ CA−1 , ఃᇏ 4 4 5 0 10 4 5 0 ⎜ 2 A = ⎝ 2 3 1 ⎠ , B = ⎝ 2 3 1 −1 ⎠ , C = ⎜ ⎝ 2 2 7 9 −3 2 7 −3 −2 −24 0 1 ⎝ − 1 0 −24 ճσ: A B = − 24 −10 −35 ⎛ −24 0 0 ⎜ 0 − 24 − 2 1 ⎜ CA−1 = − 24 ⎝ 96 −216 −18 112 −192 −22 ⎛ ⎞ 60 −175 −48 92 ⎠, −45 98 ⎞ ⎟ ⎟. ⎠ ⎛ ⎞ ⎛ ⎞ ⎛ 5 3 7 3 ⎞ 0 1 ⎟ ⎟. 9 ⎠ 7
π π 7. ഡ ω ൞ n ՑЧჰֆ໊۴ (ॖഡ ω = e2πi/n = cos 2n + i sin 2n ), ൫ Fourier इᆔ
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
1 ω n−1 ω 2(n−1) · · · ֥इᆔ.
1 1 1 . . .
1 ω ω2 . . .
1 ω2 ω4 . . .
(2) ഡइᆔ A ॖ, D − CA−1 B ္ॖ, ᆣૼॶٳइᆔ ࢳğ(1) A B 0 C
−1
=
A−1 −A−1 BC −1 . 0 C −1
(2) Ⴎ 13 ีॖᆩॶٳھइᆔॖ. ۴ऌ 13 ีᆣૼᇏ֥༯ඍ֩ൔ I 0 −CA−1 I A B C D 6 = A B 0 D − CA−1 B ,
ᄜႮЧี (1) ᇏ֥ࢲંॖᆩ (ၛ༯࠺ G = D − CA−1 B ) A B C D =
−1
=(
I 0 − 1 −CA I
−1
A B 0 G =
)−1 =
A B 0 G
−1
I 0 − 1 −CA I .
A−1 −A−1 BG−1 0 G−1
I 0 −CA−1 I
A−1 + A−1 BG−1 CA−1 −A−1 BG−1 −G−1 CA−1 G−1
4. ഡ A, B ∈ Mn , ᆣૼ: adj (AB ) = adj (B )adj (A). ᆣૼğ(1) ೂ ݔA, B नॖđᄵ֩ൔཁಖӮ৫Ġ (2) ഡ x ∈ F ູҕඔ, ᄵԢႵཋ۱ x ຓ, A − xI ა B − xI ؓఃჅཋ؟۱ඔनॖ, ၹՎ ֩ൔؓᆃཋ؟۱ඔ x नӮ৫đࠧ adj ((A − xI )(B − xI )) = adj (B − xI )adj (A − xI ). Ⴟ൞ഈൔਆ؊इᆔ֥ၩֻ i ྛֻ j ਙཌྷႋ໊ᇂ֥ჭनཌྷ֩đᆃུჭन൞ x ֥؟ཛ ൔđطਆ۱؟ཛൔೂؓݔཋ؟۱ඔनཌྷ֩đᄵૌᆺି൞၂۱؟ཛൔđၹՎഈൔਆ؊֥ ཌྷႋ໊ᇂ֥ჭؓo෮Ⴕpx नཌྷ֩đหљؓ x = 0 ္ཌྷ֩đࠧ adj (AB ) = adj (B )adj (A). 5. ᆣૼ: ؓၩइᆔ A, Ⴕ r(A∗ A) = r(AA∗ ) = r(A). ᆣૼğᆺᆣֻؽ۱֩ݼ. ഡ α ൞ٚӱ yAA∗ = 0 ֥ࢳ, ᄵ (αAA∗ )α∗ = 0đࠧ (αA)(αA)∗ = 0, Վࠧཟਈ αA ֥ଆӉູ֥ٚ 0đၹՎ αA = 0. ܣٚӱ yAA∗ = 0 აٚӱ yA = 0 ࢳđၹ Վਆ۱༢ඔइᆔ֩ᇇ. 6. ᆣૼ: ؓၩ n ࢨइᆔ A, Ⴕ r(An ) = r(An+1 ). 4
1 0 0 0 0 1 −1 −1
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18. ᆣૼֻᇕԚ֩इᆔ (ࠧ I + aEij , i = j, a = 0) дՎཌྷර. Ⴛ, ֻ၂ᇕԚ֩इᆔ൞ڎ дՎཌྷර? ᆣૼğູᆣૼֻᇕԚ֩इᆔ (ࠧ I + aEij , i = j, a = 0) дՎཌྷරđ༵ᆣૼ Eij (i = j ) न ა E12 ཌྷරࠧॖ. ູՎđ i = 1đᄵࢌߐ Eij ֻ֥ 2 ਙაֻ j ਙđಖުࢌߐྍइᆔ (Վൈྍ֥ 7
11. ഡ A ൞ n ࢨइᆔ, ؓၩ 0 = x ∈ F n नႵ Ax = x, ᆣૼ I − A ॖѩః. ᆣૼğ่ࡱ Ax = x іૼइᆔ A ֥หᆘᆴन҂֩Ⴟ 1, ၹՎ I − A ॖ. ೂ ݔA ֥௶϶ ࣥ ρ(A) < 1, ᄵ (I − A)−1 = I + A + A2 + · · · + Am + · · · (Ⴗ؊֥ࠩඔ၂ק൬৻đॖҕЧ ֻᅣ). 12. ഡ n ࢨइᆔ A ॖ, x ა y ൞ n ົਙཟਈ. ೂ( ݔA + xy ∗ )−1 ॖ, ᆣૼ ShermanMorrison ܄ൔ: A−1 xy ∗ A−1 (A + xy ∗ )−1 = A−1 − . 1 + y ∗ A−1 x ᆣૼğഈൔਆ؊ൈቐӰ A ॖ֤ A(A + xy ∗ )−1 = I − ᄜ౼ॖ֤ (A + xy ∗ )A−1 = (I − ࠧ I + xy ∗ A−1 = (I − xy ∗ A−1 −1 ) . 1 + y ∗ A−1 x xy ∗ A−1 −1 ) . 1 + y ∗ A−1 x xy ∗ A−1 . 1 + y ∗ A−1 x
10. ᆣૼྛԚ֩эߐ҂ڿэइᆔ֥ਙཟਈᆭࡗ֥ཌྟܱ༢. ᆣૼğഡइᆔ A ֥ਙཟਈ α1 , · · · , αn Ⴕཌྟܱ༢ k1 α1 + · · · + kn αn = 0. A ࣜྛݖԚ֩ эߐުэູइᆔ B . ᄵթᄝॖइᆔ P ֤ B = P A. Ⴟ൞ B ֥ਙཟਈູ P α1 , · · · , P αn . ཁ ಖႵ k1 P α1 + · · · + kn P αn = 0. ࠧྛԚ֩эߐ҂ڿэइᆔ֥ਙཟਈᆭࡗ֥ཌྟܱ༢. 5
ݔM T ΩM = Ω. ᆣૼ: (1) 2n ࢨྌइᆔ֥ಆุܒӮ၂۱ಕ, ࠧྌइᆔ֥इᆔಯ൞ྌइᆔ, ਆ۱ྌइᆔ֥Ӱࠒಯ ൞ྌइᆔ; (2) ྌޅइᆔ֥ྛਙൔनູ 1. (ิൕ: ০Ⴈॶٳइᆔ.) ᆣૼğ(1) ཁಖ. (2) Ⴎ M T ΩM = Ω ॖᆩ |M |2 = 1, Ֆ| طM | = ±1. ᆣૼ |M | = 1 ࢠđĤĤĤĤ 17. ༯ਙ۲इᆔ֥ડᇇࢳٳ: 1 2 3 0 (1) A = ⎝ 0 2 1 −1 ⎠; 1 0 2 1 ճσğ 1 0 (1) A = ⎝ 0 −1 ⎠ 1 1 ⎛ ⎞ 1 2 3 0 0 −2 −1 1 ; ⎞ 1 −1 ⎜ −1 1 ⎟ ⎟ (2) A = ⎜ ⎝ −1 −1 ⎠ 1 1 ⎛ ⎛ ⎞ ⎞ 1 −1 1 1 ⎜ −1 1 −1 −1 ⎟ ⎟. (2) A = ⎜ ⎝ −1 −1 1 1 ⎠ 1 1 −1 −1 ⎛
