矩阵理论与应用(张跃辉)习题参考解答 (上海交大).pdf

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1、⎛⎞na1nn⎜a1⎟cosxsinx11⎜⎟(1);(2);(3)⎜⎜a1⎟⎟.−sinxcosx−11⎝a1⎠a√nπnπnճσ:(1)cosnxsinnx;(2)(2)ncos4sin4;(ᇿğՎॖႮ(1)−sinnxcosnx−sinnπcosnπ44.)2.ᆣૼ:ა಩ၩnࢨٚᆔॖࢌߐइᆔс൞ՂਈइᆔλI.ᆣૼ:ഡA=(aij)ა಩ၩइᆔॖߐđᄵAაࠎЧइᆔEijॖߐ,ࠧAEij=EijA.ᇿၩAEij൞jਙູAiਙఃჅਙनູ0इᆔđEijA൞iྛູAjྛఃჅྛनູ0इᆔđႮՎॖᆩA٤ؓ࢘ჭ෍नູ0đطؓ࢘ჭ෍नડቀaii=ajjđࠧA൞Ղਈइᆔ.3.০ႨԚэ

2、ߐ౰A−1BࠣCA−1,ఃᇏ⎛⎞⎛⎞⎛⎞45045045010⎜231⎟A=⎝231⎠,B=⎝231−1⎠,C=⎜⎟.⎝279⎠27−3279−3−237⎛⎞−24060−175ճσ:A−1B=−1⎝0−24−4892⎠,24−10−35−4598⎛⎞−2400−11⎜⎜0−24−2⎟⎟CA=−.24⎝96−216−18⎠112−192−224.ഡA,B∈Mn,ᆣૼ:adj(AB)=adj(B)adj(A).ᆣૼğ(1)ೂݔA,Bनॖ୉đᄵൔཁಖӮ৫Ġ(2)ഡx∈Fູҕඔ,ᄵԢႵཋ۱؟ཋ໭ჅఃؓxI−BაxI−A,ຓx۱ඔनॖ୉,ၹՎൔ۱؟ཋ໭ᆃؓඔxनӮ৫đࠧadj((A−xI)(

3、B−xI))=adj(B−xI)adj(A−xI).Ⴟ൞ഈൔਆइᆔ಩ၩiྛjਙཌྷႋ໊ᇂჭ෍नཌྷđ֌ᆃུჭ෍न൞x؟ཛൔđਆൔഈՎၹđൔཛ؟၂۱๝൞ିᆺૌ෱ᄵđཌྷनඔ۱؟ཋ໭ؓݔೂൔཛ؟۱ਆطཌྷႋ໊ᇂჭ෍ؓç෮Ⴕéxनཌྷđหљؓx=0္ཌྷđࠧadj(AB)=adj(B)adj(A).5.ᆣૼ:ؓ಩ၩइᆔA,Ⴕr(A∗A)=r(AA∗)=r(A).ᆣૼğᆺᆣ۱.ഡα൞ٚӱyAA∗=0ࢳ,ᄵ(αAA∗)α∗=0đࠧ(αA)(αA)∗=0,ՎࠧཟਈαAଆӉ௜ູٚ0đၹՎαA=0.ܣٚӱyAA∗=0აٚӱyA=0๝ࢳđၹՎਆ۱༢ඔइᆔᇇ.6.ᆣૼ:ؓ಩ၩnࢨइᆔA,Ⴕr(An)=r(An+1)

4、.4ᆣૼğႮႿr(Ai+1)≤r(Ai),ܣA,A2,···,An+1ᇏсႵ2۱इᆔAsაAtᇇཌྷ๝.҂ٞഡs

5、n⎛⎞111···1⎜1ωω2···ωn−1⎟⎜⎟⎜⎜1ω2ω4···ω2(n−1)⎟⎟⎜....⎟⎝........⎠1ωn−1ω2(n−1)···ω(n−1)(n−1)୉इᆔ.ࢳğ࠺ีᇏइᆔູF.ᇿၩؓ಩ၩ0

6、−1a1,i+1···a1n·····················(−1)(1+i+1)−xa···aa···ajj1j,i−1j,i+1jn·····················−xnan1···an,i−1an,i+1···annࡼഈ૫ྛਙൔοjྛᅚषѩॉҳxj༢ඔđॖᆩఞູ−(−1)j+1Mij,ఃᇏMijູइᆔAჭ෍aijჅሰൔ.ၹՎ,Ց྘fᇏxixj༢ඔູ(−1)i+jMij=Aijđܣfइᆔ൞Aϴෛइᆔሇᇂđ֌ႮႿAؓӫđܣఃϴෛइᆔ္ؓӫ.9.ᆣૼइᆔᇇFrobenius҂ൔ:r(AB)+r(BC)≤r(B)+r(ABC).ᆣૼğ০Ⴈ༯ඍٳॶइᆔԚэߐࠧॖ

7、BBB−BC→→.ABCABABCAB010.ᆣૼྛԚэߐ҂ڿэइᆔਙཟਈᆭࡗཌྟܱ༢.ᆣૼğഡइᆔAਙཟਈα1,···,αnႵཌྟܱ༢k1α1+···+knαn=0.AࣜݖྛԚэߐުэູइᆔB.ᄵթᄝॖ୉इᆔP൐B=PA.Ⴟ൞BਙཟਈູPα1,···,Pαn.ཁಖႵk1Pα1+···+knPαn=0.ࠧྛԚэߐ҂ڿэइᆔਙཟਈᆭࡗཌྟܱ༢.511.ഡA൞nࢨइᆔ,ؓ಩ၩ0=x∈FnनႵAx=x,ᆣૼI−Aॖ୉ѩ౰ః୉.ᆣૼğ่ࡱAx=xіૼइ