Eigenvalues and Eigenvectors §5.2 Diagonalization

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PreviewDiagonalizationExamplesExplicit DiagonalizationEigenvalues and Eigenvectors DiagonalizationSatya Mandal, KUSummer 2017Satya Mandal, KUEigenvalues and Eigenvectors DiagonalizationPreviewDiagonalizationExa mplesExplicit DiagonalizationGoalsSupposeAis square matrix of necessary and sufficient condition when there isan invertible matrixPsuch thatP 1APis a Mandal, KUEigenvalues and Eigenvectors DiagonalizationPreviewDiagonalizationExa mplesExplicit DiagonalizationDefinitionsITwo square matricesA,Bare said to be similar, if thereis an invertible matrixP, such thatA=P square matrixAsaid to be diagonalizable, if there is aninvertible matrixP, such thatP 1APis a diagonalmatrix. That means, ifAis similar to a diagonal matrix,we say thatAis Mandal, KUEigenvalues and Eigenvectors DiagonalizationPreviewDiagonalizationExa mplesExplicit DiagonalizationTheorem ,Bare two similar matrices.

0 3 1 0 0 3 1 A: Show that A is not diagonalizable. Solution: Use Theorem 5.2.2 and show that A does not have 3 linearly independent eigenvectors. I To nd the eigenvalues, we solve det( I A) = 1 1 1 0 + 3 1 0 0 + 3 = ( 1)( +3)2 = 0: So, = 1; 3 are the only eigenvalues of A: Satya Mandal, KU Eigenvalues and Eigenvectors x5.2 Diagonalization

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