南航双语矩阵论matrix-theory第4章部分习题参考答案.docx

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第四章部分习题参考答案

Exercise4

ShowthatAandhavethesameeigenvalues.Dotheynecessarilyhavethesameeigenvectors?Explain.

SolutionSince,Aandhavethesamecharacteristicpolynomials.Hence,Aandhavethesameeigenvalues.

Aanddonotnecessarilyhavethesameeigenvectors.

Forexample,

Let,then.

TheeigenvectorsofAarethenonzerovectorsin.Theeigenvectorsofarethenonzerovectorsin.Theintersection.Hence,Aanddonothaveeigenvectorsincommon.

Exercise6

LetQbeaunitaryororthogonalmatrix.

(a)ShowthatifisaneigenvalueofQ,then

(b)Showthat|det(Q)|=1.

Proof

(a)LetxbeaneigenvectorofQcorrespondingto.Then.

hence,

(b),,.Hence,|det(Q)|=1.

Exercise9

LetQbeaorthogonalmatrixwhosedeterminantisequalto1.

(a)IftheeigenvalueofQareallrealandiftheyareorderedsothat,determinethevaluesofallpossibletriplesofeigenvalues

(b)Inthecasethattheeigenvaluesandarecomplex,whatarethepossiblevaluesfor?Explain.

Solution

ByExercise#6,for.IftheeigenvalueofQareallreal,then.

Since,.Hence,thepossibletriplesofeigenvalueswithare

.

Iftheeigenvaluesandarecomplex,thenandmustbetheconjugateofeachother..Hence,since.

Exercise11

LetbeanorthonormalbasisforandletAbealinearcombinationofrank1matrices,,…,.If

ShowthatAisasymmetricmatrixwitheigenvaluesandthatisaneigenvectorbelongingtoforeachi.

Proof

Hence,Aisasymmetricmatrix.

Hence,isaneigenvectorbelongingtoforeachi.

Exercise13

LetAandBbematrices.Showthat

IfisanonzeroeigenvalueofAB,thenitisalsoaneigenvalueofBA.

IfisaneigenvalueofAB,thenisalsoaneigenvalueofBA.

Proof

IfisanonzeroeigenvalueofAB,thenthereisanonzerovectorxsuchthat

.From,weseethat.

Since,weobtainthatisaneigenvectorofBAcorrespondingtotheeigenvalue.Hence,is