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# 338 Class Note for MATH 220 at PSU

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This 1 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 28 views.

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Date Created: 02/06/15
MATH 220 Similar Matrices and Diagonalization 0 De nition of Similar Matrices Two 71 by n matrices A and B are said to be similar if there exists an invertible n by 71 matrix C such that 304AO D The function de ned by equation 1 that takes the matrix A and returns the matrix B is called a similarity transformation 0 Theorem 1 from 63 If A and B are similar 71 by n matrices then A and B have the same characteristic polynomial and therefore have the same eigenvalues 0 De nition of Diagonalizable An 71 by 71 matrix is diagonalizable if there is a diagonal matrix D such that A is similar to D 0 Theorem 2 from 63 An 71 by 71 matrix A is diagonalizable if and only if it has n linearly independent eigenvectors In that case the diagonal matrix D similar to A is A1 0 0 0 0 A2 0 0 D 0 0 A3 0 0 0 0 An where A1 A2 A are the eigenvalues of A Furthermore if C is a matrix whose columns are linearly independent eigenvectors of A then D04AO 4 2 0 Example For A lt 3 3 gt nd 0 which diagonalizes A by equation Verify that D C lAC 71 a b i 1 d 7b note lt C d 7 71470 lt 76 a gt If the n by 71 matrix A has n distinct eigenvalues then A is diagonalizable 0 Corollary from 63 0 Purpose of 64 Q What kind of matrices can be diagonalized A For starters real symmetric n by n matrices have n linearly independent eigenvectors and so they can be diagonalized

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