线性代数及其应用 英文版 答案 第五章

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1、5.1SOLUTIONSNotes:Exercises1–6reinforcethedefinitionsofeigenvaluesandeigenvectors.Thesubsectiononeigenvectorsanddifferenceequations,alongwithExercises33and34,referstothechapterintroductoryexampleandanticipatesdiscussionsofdynamicalsystemsinSections5.2and5.6.1.Thenumber2isaneigenvalueofAifand

2、onlyiftheequationAxx=2hasanontrivialsolution.Thisequationisequivalentto(2AI−=).x0Compute322012AI−=2−=380236ThecolumnsofAareobviouslylinearlydependent,so(AI−=2)x0hasanontrivialsolution,andso2isaneigenvalueofA.2.Thenumber2−isaneigenvalueofAifandonlyiftheequationAxx=−2hasanont

3、rivialsolution.Thisequationisequivalentto(AI+=2)x0.Compute732093AI+=2+=310231−ThecolumnsofAareobviouslylinearlydependent,so(AI+=2)x0hasanontrivialsolution,andso−2isaneigenvalueofA.−3111113.IsAxamultipleofx?Compute=≠λ.SoisnotaneigenvectorofA.−38429

4、4421−+12−+1224.IsAxamultipleofx?Compute=ThesecondentriesofxandAxshows14132+thatifAxisamultipleofx,thenthatmultiplemustbe32+.Check32+timesthefirstentryofx:2(3+−2)(1+=2)−3++=222−122+−+12ThismatchesthefirstentryofAx,soisaneigenvectorofA,andthecorresponding1

5、eigenvalueis32+.257258CHAPTER5•EigenvaluesandEigenvectors3794045.IsAxamultipleofx?Compute−−45130−=.So−3isaneigenvectorofAforthe244101eigenvalue0.3671−2116.IsAxamultipleofx?Compute3372−=4(2=−−)2So−2isaneigenvectorof5651

6、−211Afortheeigenvalue−2.7.Todetermineif4isaneigenvalueofA,decideifthematrixAI−4isinvertible.301400−−101−AI−=4231040−=211−−−345004341Invertibilitycanbecheckedinseveralways,butsinceaneigenvectorisneededintheeventthatoneexists,thebeststrategyistoro

7、wreducetheaugmentedmatrixfor(AI−=4)x0:−−−−101010101010211001100110−−∼∼−−−−341004400000Theequation(AI−=4)x0hasanontrivialsolution,so4isaneigenvalue.Anynonzerosolutionof(4AI−=)x0isacorrespondingeigenvector.Theentriesinas