南航双语矩阵论matrix-theory第4章部分习题参考答案

第四章部分习题参考答案Exercise4ShowthatAandhavethesameeigenvalues.Dotheynecessarilyhavethesameeigenvectors?Explain.SolutionSince,Aandhavethesamecharacteristicpolynomials.Hence,Aandhavethesameeigenvalues.Aanddonotnecessarilyhavethesameeigenvectors.Forexample,Let,then.TheeigenvectorsofAarethenonzerovectorsin.Theeigenvectorsofarethenonzerovectorsin.Theintersection.Hence,Aanddonothaveeigenvectorsincommon.

Exercise6LetQbeaunitaryororthogonalmatrix.

(a)ShowthatifisaneigenvalueofQ,then(b)Showthat|det(Q)|=1.

Proof(a)LetxbeaneigenvectorofQcorrespondingto.Then.hence,(b),,.Hence,|det(Q)|=1.Exercise9LetQbeaorthogonalmatrixwhosedeterminantisequalto1.(a)IftheeigenvalueofQareallrealandiftheyareorderedsothat,determinethevaluesofallpossibletriplesofeigenvalues(b)Inthecasethattheeigenvaluesandarecomplex,whatarethepossiblevaluesfor?Explain.SolutionByExercise#6,for.IftheeigenvalueofQareallreal,then.Since,.Hence,thepossibletriplesofeigenvalueswithare.Iftheeigenvaluesandarecomplex,thenandmustbetheconjugateofeachother..Hence,since.Exercise11LetbeanorthonormalbasisforandletAbealinearcombinationofrank1matrices,,…,.IfShowthatAisasymmetricmatrixwitheigenvaluesandthatisaneigenvectorbelongingtoforeachi.

ProofHence,Aisasymmetricmatrix.Hence,isaneigenvectorbelongingtoforeachi.

Exercise13LetAandBbematrices.ShowthatIfisanonzeroeigenvalueofAB,thenitisalsoaneigenvalueofBA.IfisaneigenvalueofAB,thenisalsoaneigenvalueofBA.ProofIfisanonzeroeigenvalueofAB,thenthereisanonzerovectorxsuchthat.From,weseethat.Since,weobtainthatisaneigenvectorofBAcorrespondingtotheeigenvalue.Hence,isalsoaneigenvalueofBA.IfisaneigenvalueofAB,then.Hence,.Thus,isaneigenvalueofBA.Exercise14ProvethattheredonotexistmatricesAandBsuchthat.ProofLetand.,.Hence,.,ItisimpossibletohavematricesAandBsuchthat.Exercise15Letbeapolynomialofdegree,andlet(ThematrixCiscalledthecompanionmatrixof.)(a)Showthatifisarootofthenisaneigenvalueofwitheigenvector.(b)Usepart(a)toshowthatifhasndistinctrootsthenisthecharacteristicpolynomialofC.(Theresultistrueevenifalltheeigenvaluesofarenotdistinct.)Hint:Candhavethesamecharacteristicpolynomial.ProofIfisarootof,then.Weobtainthat.Thenisaneigenvalueofwitheigenvector.Ifhasndistinctroots,thenallrootsofareeigenvaluesof.Weobtainthatthecharacteristicpolynomialofandhavethesamendistinctroots.Andalsotheyhavethesamedegreeandthesameleadingcoefficient.Hence,thecharacteristicpolynomialofisthesameas.SinceCandhavethesamecharacteristicpolynomial,weknowthatisthecharacteristicpolynomialofC.Exercise16LetbeanorthogonaltransformationonaEuclideanspaceV(aninnerproductspaceovertherealnumberfield).IfWisa-invariantsubspaceofV,showthattheorthogonalcomplementofWisalso-invariant.ProofLet,whereWis-invariant.LetbeanorthonormalbasisforW.ThenisalsoanorthonormalbasisforWsinceWis-invariantandisanorthogonaltransformation.(Anorthogonaltransformationmapsanorthonormalsettoanorthonormalset.)Let.Forany,ucanbewrittenasalinearcombinationof.Sinceorthogonaltransform