A9_soln - Math 235 Assignment 9 Solutions 1. Suppose that a...

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Math 235 Assignment 9 Solutions 1. Suppose that a real 2 × 2 matrix A has 2 + i as an eigenvalue with a corresponding eigenvector ± 1 + i i ² . Determine A . Solution: Since A is real, we know that A has real canonical form B = ± 2 1 - 1 2 ² and is brought into this form by P = ± 1 1 0 1 ² . We then have that A = PBP - 1 = ± 1 1 0 1 ²± 2 1 - 1 2 ²± 1 - 1 0 1 ² = ± 1 2 - 1 3 ² . 2. Determine a real canonical form of A = 0 - 2 1 2 2 - 1 0 - 2 2 and give a change of basis matrix P that brings the matrix into this form. Solution: We have C ( λ ) = - ( λ - 2)( λ 2 - 2 λ + 2). The roots are 2 and 1 ± i , so the eigenvalues of A are μ = 2, λ = 1 + i and λ . Hence, a real canonical form is 2 0 0 0 1 1 0 - 1 1 . For μ = 2 we get A - μI = - 2 - 2 1 2 0 - 1 0 - 2 0 1 0 - 1 / 2 0 1 0 0 0 0 . So, an eigenvector corresponding to μ is ~v = 1 0 2 . For λ = 1 + i we get A - λI = - 1 - i - 2 1 2 1 - i - 1 0 - 2 1 - i 1 0 - 1 2 - 1 2 i 0 1 - 1 2 + 1 2 i 0 0 0 . So, an eigenvector corresponding to
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This note was uploaded on 11/30/2010 for the course MATH 235/237 taught by Professor Wilkie during the Spring '10 term at Waterloo.

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A9_soln - Math 235 Assignment 9 Solutions 1. Suppose that a...

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