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Eigenvalues and Eigenvectors §5.2 Diagonalization

PreviewDiagonalizationExamplesExplicit DiagonalizationEigenvalues and Eigenvectors DiagonalizationSatya Mandal, KUSummer 2017 Satya Mandal, KUEigenvalues and Eigenvectors DiagonalizationPreviewDiagonalizationExa mplesExplicit DiagonalizationGoalsSupposeAis square matrix of necessary and sufficient condition when there isan invertible matrixPsuch thatP 1 APis 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 1 APis 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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