Transcription of Generalized Eigenvectors - University of Pennsylvania
1 GeneralizedEigenvectorsMath 240 DefinitionComputationand PropertiesChainsGeneralized EigenvectorsMath 240 Calculus IIIS ummer 2013, Session IIWednesday, July 31, 2013 GeneralizedEigenvectorsMath 240 DefinitionComputationand PropertiesChainsAgenda1. Definition2. Computation and Properties3. ChainsGeneralizedEigenvectorsMath 240 DefinitionComputationand PropertiesChainsMotivationDefective matrices cannot be diagonalized because they do notpossess enough Eigenvectors to make a basis. How can wecorrect this defect?ExampleThe matrixA=[1 10 1]is Only eigenvalue is = I=[0 10 0]3. Single eigenvectorv= (1,0).4. We could useu= (0,1)to complete a Notice that(A I)u=vand(A I)2u= we just didn t multiply byA Ienough 240 DefinitionComputationand PropertiesChainsDefinitionDefinitionIfAi s ann nmatrix, ageneralized eigenvectorofAcorresponding to the eigenvalue is a nonzero vectorxsatisfying(A I)px=0for some positive , it is a nonzeroelement of the nullspace of(A I) are Generalized Eigenvectors withp= the previous example we saw thatv= (1,0)andu= (0,1)are Generalized Eigenvectors forA=[1 10 1]and = 240 DefinitionComputationand PropertiesChainsComputing Generalized eigenvectorsExampleDetermine Generalized Eigenvectors for the matrixA= 1 1 00 1 20 0 3.
2 1. Characteristic polynomial is(3 )(1 ) Eigenvalues are = 1, Eigenvectors are 1= 3 :v1= (1,2,2), 2= 1 :v2= (1,0,0).4. Final Generalized eigenvector will a vectorv36=0suchthat(A 2I)2v3=0but(A 2I)v36= (0,1,0).Note that(A 2I)v3= 240 DefinitionComputationand PropertiesChainsFacts about Generalized eigenvectorsHow many powers of(A I)do we need to compute in orderto find all of the Generalized Eigenvectors for ?FactIfAis ann nmatrix and is an eigenvalue with algebraicmultiplicityk, then the set of Generalized Eigenvectors for consists of the nonzero elements ofnullspace((A I)k).In other words, we need to take at mostkpowers ofA Itofind all of the Generalized Eigenvectors for .GeneralizedEigenvectorsMath 240 DefinitionComputationand PropertiesChainsComputing Generalized eigenvectorsExampleDetermine Generalized Eigenvectors for the matrixA= 1 2 01 1 20 1 1 .1. Single eigenvalue of = Single eigenvectorv1= ( 2,0,1).3. Look at(A I)2= 2 0 40 0 0 1 0 2 to find Generalized eigenvectorv2= (0,1,0).
3 4. Finally,(A I)3=0, so we getv3= (1,0,0).GeneralizedEigenvectorsMath 240 DefinitionComputationand PropertiesChainsFacts about Generalized eigenvectorsThe aim of Generalized Eigenvectors was to enlarge a set oflinearly independent Eigenvectors to make a basis. Are therealways enough Generalized Eigenvectors to do so?FactIf is an eigenvalue ofAwith algebraic multiplicityk, thennullity((A I)k)= other words, there areklinearly independent generalizedeigenvectors for .CorollaryIfAis ann nmatrix, then there is a basis forRnconsistingof Generalized Eigenvectors 240 DefinitionComputationand PropertiesChainsComputing Generalized eigenvectorsExampleDetermine Generalized Eigenvectors for the matrixA= 1 2 01 1 20 1 1 .1. From last time, we have eigenvalue = 1and eigenvectorv1= ( 2,0,1).2. Solve(A I)v2=v1to getv2= (0, 1,0).3. Solve(A I)v3=v2to getv3= ( 1,0,0).GeneralizedEigenvectorsMath 240 DefinitionComputationand PropertiesChainsChains of Generalized eigenvectorsLetAbe ann nmatrix andva Generalized eigenvector ofAcorresponding to the eigenvalue.
4 This means that(A I)pv=0for a positive q < p, then(A I)p q(A I)qv= is,(A I)qvis also a Generalized eigenvectorcorresponding to forq= 0,1,..,p the smallest positive integer such that(A I)pv=0,then the sequence(A I)p 1v,(A I)p 2v, ..,(A I)v,vis called achainorcycleof Generalized Eigenvectors . Theintegerpis called thelengthof the 240 DefinitionComputationand PropertiesChainsChains of Generalized eigenvectorsExampleIn the previous example,A I= 0 2 01 0 20 1 0 and we found the chainv= 100 ,(A I)v= 0 10 ,(A I)2v= 201 .FactThe Generalized Eigenvectors in a chain are 240 DefinitionComputationand PropertiesChainsJordan canonical formWhat s the analogue of diagonalization for defective matrices?That is, if{v1,v2,..,vn}are the linearly independentgeneralized Eigenvectors ofA, what does the matrixS 1 ASlook like, whereS=[v1v2 vn]?
