资源描述：
《eigenvalues and eigenvectors特征值和特征向量》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、MAC2103Module12EigenvaluesandEigenvectors1Rev.F09LearningObjectivesUponcompletingthismodule,youshouldbeableto:Solvetheeigenvalueproblembyfindingtheeigenvaluesandthecorrespondingeigenvectorsofannxnmatrix.Findthealgebraicmultiplicityandthegeometricmultiplicityofaneigenvalue.F
2、indabasisforeacheigenspaceofaneigenvalue.DeterminewhetheramatrixAisdiagonalizable.FindamatrixP,P-1,andDthatdiagonalizeAifAisdiagonalizable.FindanorthogonalmatrixPwithP-1=PTandDthatdiagonalizeAifAissymmetricanddiagonalizable.Determinethepowerandtheeigenvaluesofamatrix,Ak.htt
3、p://faculty.valenciacc.edu/ashaw/Clicklinktodownloadothermodules.2Rev.09EigenvaluesandEigenvectorshttp://faculty.valenciacc.edu/ashaw/Clicklinktodownloadothermodules.Eigenvalues,Eigenvectors,Eigenspace,DiagonalizationandOrthogonalDiagonalizationThemajortopicsinthismodule:3R
4、ev.F09WhatareEigenvaluesandEigenvectors?http://faculty.valenciacc.edu/ashaw/Clicklinktodownloadothermodules.IfAisannxnmatrixandλisascalarforwhichAx=λxhasanontrivialsolutionx∈ℜⁿ,thenλisaneigenvalueofAandxisacorrespondingeigenvectorofA.Ax=λxiscalledtheeigenvalueproblemforA.No
5、tethatwecanrewritetheequationAx=λx=λInxasfollows:λInx-Ax=0or(λIn-A)x=0.x=0isthetrivialsolution.Butoursolutionsmustbenonzerovectorscalledeigenvectorsthatcorrespondtoeachofthedistincteigenvalues.4Rev.F09WhatareEigenvaluesandEigenvectors?http://faculty.valenciacc.edu/ashaw/Cli
6、cklinktodownloadothermodules.Sinceweseekanontrivialsolutionto(λIn-A)x=(λI-A)x=0,λI-Amustbesingulartohavesolutionsx≠0.Thismeansthatthedet(λI-A)=0.Thedet(λI-A)=p(λ)=0isthecharacteristicequation,wheredet(λI-A)=p(λ)isthecharacteristicpolynomial.Thedeg(p(λ))=nandthenrootsofp(λ),
7、λ1,λ2,…,λn,aretheeigenvaluesofA.Thepolynomialp(λ)alwayshasnroots,sothezerosalwaysexist;butsomemaybecomplexandsomemayberepeated.Inourexamples,alloftherootswillbereal.Foreachλiwesolveforxi=pithecorrespondingeigenvector,andApi=λipiforeachdistincteigenvalue.5Rev.F09HowtoSolveth
8、eEigenvalueProblem,Ax=λx?http://faculty.valenciacc.edu/ashaw/Clicklinktodownloadot
