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《南航双语矩阵论matrix theory第7章部分习题参考答案》由会员上传分享，免费在线阅读，更多相关内容在教育资源-天天文库。
1、第七章部分习题参考答案Exercise1ShowthatanormalmatrixAisHermitianifitseigenvaluesareallreal.ProofIfAisanormalmatrix,thenthereisaunitarymatrixthatdiagonalizesA.Thatis,thereisaunitarymatrixUsuchthatwhereDisadiagonalmatrixandthediagonalelementsofDareeigenvaluesofA.
2、IfeigenvaluesofAareallreal,thenTherefore,AisHermitian.Exercise2LetAandBbeHermitianmatricesofthesameorder.ShowthatABisHermitianifandonlyif.ProofIf,then.Hence,ABisHermitian.Conversely,ifABisHermitian,then.Therefore,.Exercise3LetAandBbeHermitianmatrices
3、ofthesameorder.ShowthatAandBaresimilariftheyhavethesamecharacteristicpolynomial.ProofSincematrixAandBhavethesamecharacteristicpolynomial,theyhavethesameeigenvalues.ThereexistunitarymatricesUandVsuchthat,.Thus,.()Thatis.Hence,AandBaresimilar.Exercise4
4、LetAbeaskew-Hermitianmatrix,i.e.,,showthat(a)andareinvertible.(b)isaunitarymatrixwitheigenvaluesnotequalto.ProofofPart(a)Method1:(a)since,itfollowsthatForanyHence,ispositivedefinite.Itfollowsthatisinvertible.Hence,bothandareinvertible.Method2:5Ifissi
5、ngular,thenthereexistsanonzerovectorxsuchthat.Thus,,.(1)Sinceisreal,itfollowsthat.Thatis.Since,itfollowsthat(2)Equation(1)and(2)impliesthat.Thiscontradictstheassumptionthatxisnonzero.Therefore,isinvertible.Method3:LetbeaneigenvalueofAandxbeanassociat
6、edeigenvector..Hence,iseitherzeroorpureimaginary.1andcannotbeeigenvaluesofA.Hence,andareinvertible.Method4:Since,Aisnormal.ThereexistsaunitarymatrixUsuchthatEachispureimaginaryorzero.Sincefor,det.Hence,isinvertible.Similarly,wecanprovethatisinvertibl
7、e.ProofofPart(b)Method1:Since,itfollowsthat(NotethatifPisnonsingular.)Hence,isaunitarymatrix.Denote.Since,Hence,cannotbeaneigenvalueof.Method2:Bymethod4oftheProofofPart(a),5Theeigenvaluesofare,whichareallnotequalto.Method3:Since,itfollowsthatIfisanei
8、genvalueof,thenthereisanonzerovectorx,suchthat.Thatis.Itfollowsthat.Thisimpliesthat.Thiscontradictionshowsthatcannotbeaneigenvalueof.Exercise6IfHisHermitian,showthatisinvertible,andisunitary.ProofLet.ThenAisskew-Hermitian.ByExercises#4,andareinvertib
