线性代数英文课件：ch5 1Eigenvalues and Eigenvectors

Chapter5SimilarMatricesandQuadraticForms

Sec.1EigenvaluesandEigenvectors

Sec.2SimilarMatrices

Sec.3TheSimilarMatricesofRealSymmetricMatrices

Chapter5SimilarMatrices1Inthischapter,thefollowingproblemswillbediscussedmainly:TheConceptsofEigenvalueandEigenvector;

TheNecessary&SufficientConditionsforaMatrixtobeDiagonalizable;Inthischapter,thefollo2Sec.1EigenvaluesandEigenvectors1.DefinitionofEigenvaluesandEigenvectors3.PropertisofEigenvaluesandEigenvectors2.HowtofindtheEigenvaluesandEigenvectorsofamatrix?4.ReviewSec.1EigenvaluesandEigenvec3EigenvaluesandEigenvectorsAxForexample,EigenvalueEigenvaluesandEigenvectors4?Howmanyeigenvaluesdoesamatrixhave?Twoeigenvalues:5,-1?Howmanyeigenvaluesdoesam5?Howtofindoutalltheeigenvalues?TheeigenvectorofAcorrespondingtotheeigenvalueisthenontrivialsolutionofhomogenousLS:Why??Howtofindoutalltheeigen6Definition2Characteristicpolynomial

（特征多项式）

Characteristicequation（特征方程）:TheeigenvaluesofmatrixAaretherootsofthecharacteristicpolynomialofA.HowmanyeigenvaluesdoesamatrixAhave??Definition2Characteristicpol7SotheeigenvaluesofAis:Theeigenvaluesofadiagonalmatrixaretheelementsinthediagonals(对角矩阵的特征值为对角线上的元素)SotheeigenvaluesofAis:The82.HowtofindtheEigenvaluesandEigenvectorsofamatrix?TwoSteps:TofindouttheeigenvaluesofA;Tofindouttheeigenvectorscorrespondinglytoλ,whichisthenontrivialsolutionsofthisLS.Eigenvectorisanonzerovector！2.HowtofindtheEigenvalues9Solution:WhenSolution:When10线性代数英文课件：ch51EigenvaluesandEigenvectors11Solution:WhenSolution:When12线性代数英文课件：ch51EigenvaluesandEigenvectors13TheequivalentLSisTheequivalentLSis143.PropertisofEigenvaluesandEigenvectorsProperty1特征值之和=对角线元素之和；特征值之积=方阵的行列式3.PropertisofEigenvaluesand15ifAmusthasatleastonezeroeigenvalue.ifAmusthasnonzeroeigenvalue.Howmanyconclusionscanwedraw?ifAmusthasatleaston16LetbeaneigenvalueofmatrixA,thenisaneigenvalueofmatrixAn.

Property2Proof:Furthermore,Letbeaneigenvalueofmatrix17Example3.Aisamatrixoforder3,and|A-E|=0,|A+3E|=0,|A-4E|=0.FindtheeigenvaluesofA;Find|A|;FindtheeigenvaluesofA2+A+2E;Find|A2+A+2E|Solution:(2)|A|=1*(-3)*4=-12;(1)TheeigenvaluesofAare1,-3,4;(3)TheeigenvaluesofA2+A+2Eis4,8,22.（4）|A2+A+2E|＝4×8×22＝704Example3.Aisamatrixofor18Property3(i)ThetransposeofAhavethesameeigenvalueswithA.Methodstoprovethatmatriceshavethesameeigenvalues.Property3(i)Thetransposeof19Supposethereexisttwonumbersk1,k2,suchthatProperty4Proof:|Thenk2=0Supposethereexisttwonumber20Furthermore,wehave:Inaword,Theeigenvectorsaccordingtodifferenteigenvaluesofamatrixislinearlyindependent.（不同的特征值所对应的特征向量线性无关）Furthermore,wehave:Inaword214.Review(1)HowtofindtheEigenvaluesandEigenvectorsofamatrix?TwoSteps.(2)ifλisaneigenvalueofmatrixA,(3)Whatistheeigenvaluesofadiagonalmatrix?4.Review(1)HowtofindtheEi22ThescalarλandvectorxarerespectivelytheeigenvalueandcorrespondingeigenvectorofsquarematrixAiftheysatisfyAx=λx.(2)IfλandμareeigenvaluesofmatrixAandBrespectively,thenλ+μareeigenvalueofA+B.Judgethefollowingstatements:××Thescalarλandvectorxare23Chapter5SimilarMatricesandQuadraticForms

Sec.1EigenvaluesandEigenvectors

Sec.2SimilarMatrices

Sec.3TheSimilarMatricesofRealSymmetricMatrices

Chapter5SimilarMatrices24Inthischapter,thefollowingproblemswillbediscussedmainly:TheConceptsofEigenvalueandEigenvector;

TheNecessary&SufficientConditionsforaMatrixtobeDiagonalizable;Inthischapter,thefollo25Sec.1EigenvaluesandEigenvectors1.DefinitionofEigenvaluesandEigenvectors3.PropertisofEigenvaluesandEigenvectors2.HowtofindtheEigenvaluesandEigenvectorsofamatrix?4.ReviewSec.1EigenvaluesandEigenvec26EigenvaluesandEigenvectorsAxForexample,EigenvalueEigenvaluesandEigenvectors27?Howmanyeigenvaluesdoesamatrixhave?Twoeigenvalues:5,-1?Howmanyeigenvaluesdoesam28?Howtofindoutalltheeigenvalues?TheeigenvectorofAcorrespondingtotheeigenvalueisthenontrivialsolutionofhomogenousLS:Why??Howtofindoutalltheeigen29Definition2Characteristicpolynomial

（特征多项式）

Characteristicequation（特征方程）:TheeigenvaluesofmatrixAaretherootsofthecharacteristicpolynomialofA.HowmanyeigenvaluesdoesamatrixAhave??Definition2Characteristicpol30SotheeigenvaluesofAis:Theeigenvaluesofadiagonalmatrixaretheelementsinthediagonals(对角矩阵的特征值为对角线上的元素)SotheeigenvaluesofAis:The312.HowtofindtheEigenvaluesandEigenvectorsofamatrix?TwoSteps:TofindouttheeigenvaluesofA;Tofindouttheeigenvectorscorrespondinglytoλ,whichisthenontrivialsolutionsofthisLS.Eigenvectorisanonzerovector！2.HowtofindtheEigenvalues32Solution:WhenSolution:When33线性代数英文课件：ch51EigenvaluesandEigenvectors34Solution:WhenSolution:When35线性代数英文课件：ch51EigenvaluesandEigenvectors36TheequivalentLSisTheequivalentLSis373.PropertisofEigenvaluesandEigenvectorsProperty1特征值之和=对角线元素之和；特征值之积=方阵的行列式3.PropertisofEigenvaluesand38ifAmusthasatleastonezeroeigenvalue.ifAmusthasnonzeroeigenvalue.Howmanyconclusionscanwedraw?ifAmusthasatleaston39LetbeaneigenvalueofmatrixA,thenisaneigenvalueofmatrixAn.

Property2Proof:Furthermore,Letbeaneigenvalueofmatrix40Example3.Aisamatrixoforder3,and|A-E|=0,|A+3E|=0,|A-4E|=0.FindtheeigenvaluesofA;Find|A|;FindtheeigenvaluesofA2+A+2E;Find|A2+A+2E|Solution:(2)|A|=1*(-3)*4=-12;(1)TheeigenvaluesofAare1,-3,4;(3)TheeigenvaluesofA2+A+2Eis4,8,22.（4）|A2+A+2E|＝4×8×22＝704Example3.Aisamatrixofor41Property3(i)ThetransposeofAhavethesameeigenvalueswithA.Methodstoprovethatmatriceshavethesameeigenvalues.Property3(i)Thetransposeof42Supposethereexisttwonumbersk1,k2,suchthatProperty4Proof:|Thenk2=0