A6_soln - Math 235 Assignment 6 Solutions 1. For each of...

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Math 235 Assignment 6 Solutions 1. For each of the following symmetric matrices, find the orthogonal matrix R that diagonalizes the given matrix and the corresponding diagonal matrix. a) A = ± 5 2 2 2 ² Solution: We have A - λI = ± 5 - λ 2 2 2 - λ ² which gives c ( λ ) = ( λ - 1)( λ - 6). Hence, the eigenvalues are λ = 1 and λ = 6. For λ = 1 we get A - λI = ± 4 2 2 1 ² ± 1 1 / 2 0 0 ² so a corresponding eigenvector is ~v 1 = ± 1 - 2 ² . For λ = 6 we get A - λI = ± - 1 2 2 - 4 ² ± 1 - 2 0 0 ² so a corresponding eigenvector is ~v 2 = ± 2 1 ² . Normalizing we get ˆ v 1 = 1 5 ± 1 - 2 ² and ˆ v 2 = 1 5 ± 2 1 ² . Thus R = 1 5 ± 1 2 - 2 1 ² and D = ± 1 0 0 6 ² . b) B = 1 2 - 4 2 - 2 - 2 - 4 - 2 1 Solution: We have B - λI = 1 - λ 2 - 4 2 - 2 - λ - 2 - 4 - 2 1 - λ which gives c ( λ ) = - ( λ +3) 2 ( λ - 6). Hence, the eigenvalues are λ = 6 and λ = - 3. λ = 6 gives B - λI = - 5 2 - 4 2 - 8 - 2 - 4 - 2 - 5 1 0 1 0 1 1 / 2 0 0 0 so an eigenvector is ~v 1 = 2 1 - 2 . λ = - 3 gives B - λI = 4 2 - 4 2 1 - 2 - 4 - 2 4 1 1 / 2 - 1 0 0 0 0 0 0 so two corresponding eigenvectors are ~v 2 = - 1 2 0 ,~v 3 = 1 0 1 . These are not mutually orthogonal, so we apply the Gram- Schmidt procedure to get ~w 1 = ~v 2 and ~w 2 = (1 , 0 , 1) - (1 , 0 , 1) · ( - 1 , 2 , 0) 5 ( - 1 , 2 , 0) = ( 4 5 , 2 5 , 1) . Normalizing the vectors we get ˆ
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This note was uploaded on 04/18/2010 for the course MATH 235 taught by Professor Celmin during the Spring '08 term at Waterloo.

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A6_soln - Math 235 Assignment 6 Solutions 1. For each of...

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