A9 - 6 = 0 is an eigenvalue of A . Let ~ z = ~x + i~ y be...

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Math 235 Assignment 9 Due: Wednesday, July 21st 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 . 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 the bring the matrix into this form. 3. Let V be a vector space over C and T : V V a linear operator. Prove that Null( T ) and Range( T ) are invariant subspaces of T . 4. Suppose that A is an n × n matrix with real entries, and that λ = a + bi , b
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Unformatted text preview: 6 = 0 is an eigenvalue of A . Let ~ z = ~x + i~ y be an eigenvector of A corresponding to . Prove that ~x 6 = ~ 0, ~ y 6 = ~ 0 and that ~x 6 = k~ y . 5. Suppose that A is an n n matrix with real entries, and that = a + bi , b 6 = 0 is an eigenvalue of A . Let ~ z = ~x + i~ y be an eigenvector of A corresponding to . Prove that Span { ~x,~ y } contains no real eigenvector of A . 1...
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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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