practiceprob7_soln

practiceprob7_soln - Math 235 Assignment 7 Practice...

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Assignment 7 Practice Problems Solutions 1. For each of the following symmetric matrices, find the orthogonal matrix P 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) A = 2 - 1 - 1 - 1 2 - 1 - 1 - 1 2 Solution: Observe that the eigenvalues of A are 3, 3, and 0. For λ 1 = 3 we have A - λ 1 I = - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 1 1 1 0 0 0 0 0 0 Hence, we have corresponding eigenvectors ~w 1 = - 1 1 0 and ~w 2 = - 1 0 1 . Since these are not orthgonal, we need to apply the Gram-Schmidt procedure to get ~v 1 = - 1 1 0 and ~v 2 = - 1 0 1
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practiceprob7_soln - Math 235 Assignment 7 Practice...

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