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Generalized Eigenvectors - University of Pennsylvania

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.

Jordan canonical form What’s the analogue of diagonalization for defective matrices? That is, if fv 1;v 2;:::;v ngare the linearly independent generalized eigenvectors of A, what does the matrix S 1AS look like, where S= v 1 v 2 v n?

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