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Unformatted text preview: Math 235 Assignment 6 Solutions 1. For each of the following symmetric matrices, find the orthogonal matrix R that diago nalizes the given matrix and the corresponding diagonal matrix. a) A = 7 3 3 1 Solution: The characteristic polynomails is C ( λ ) = det( A λI ) = det 7 λ 3 3 1 λ = ( λ + 2)( λ 8) . Hence, the eigenvalues of A are λ 1 = 2 and λ 2 = 8. For λ 1 = 2 we have A λ 1 I = 9 3 3 1 ∼ 1 1 / 3 . Thus, an eigenvector corresonding to λ 1 is 1 3 . For λ 2 = 8 we have A λ 2 I = 1 3 3 9 ∼ 1 3 . Thus, an eigenvector corresonding to λ 2 is 3 1 . Normalizing the eigenvectors, we find that A is diagonalied by the orthogonal matrix P = 1 √ 10 1 3 3 1 and P T AP = 2 0 8 = D. 2 b) B = 1 1 1 1 1 1 Solution: The characteristic polynomails is C ( λ ) = det( B λI ) = det  λ 1 1 1 λ 1 1 1 λ = det 1 λ 1 1 1 λ λ 1 1 λ = det 1 λ 1 1 1 λ 2 1 λ = ( λ 1)( λ 2 + λ 2) = ( λ 1) 2 ( λ + 2) . Hence, the eigenvalues of B are λ 1 = 1 and λ 2 = 2. For λ 1 = 1 we have B λ 1 I =  1 1 1 1 1 1 1 1 1 ∼  1 1 1 . Thus, linearly independent eigenvectors corresonding to λ 1 are ~v 1 = 1 1 , and ~v 2 =  1 1 . For λ 2 = 2 we have B λ 2 I = 2 1 1 1 2 1 1 1 2 ∼ 1 0 1 0 1 1 0 0 . Thus, an eigenvector corresonding to λ 2 is 1 1 1 . We first observe that eigenvectors for λ 1 are not orthogonal. So, applying the GramSchmidt procedure to { ~v 1 ,~v 2 } : Let ~w 1 = (1 , 1 , 0); then ~w 2 = ( 1 , , 1) ( 1 , ,...
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This note was uploaded on 10/12/2010 for the course MATH 235 taught by Professor Celmin during the Fall '08 term at Waterloo.
 Fall '08
 CELMIN
 Matrices

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