math 235 hw8 - Homework 8, Due May 4 Adam Gamzon Related...

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Unformatted text preview: Homework 8, Due May 4 Adam Gamzon Related reading: sections 7.5, 5.1, 5.2, 5.5 from the text 1. (a) Find all complex eigenvalues of the matrix A = 1 5- 5 10 . Solution: The characteristic polynomial is det t- 1- 5 5 t- 10 = ( t- 1)( t- 10) + 25 = t 2- 11 t + 35 . Therefore, the eigenvalues are 11 ± √ 121- 140 2 = 11 2 ± 1 2 √- 19 = 11 2 ± i √ 19 2 . (b) Find a basis of each eigenspace of A . Solution: For E 11 / 2+ i √ 19 / 2 , find ker 9 / 2 + i √ 19 / 2- 5 5- 9 / 2 + i √ 19 / 2 . To do this, solve the system 9 / 2 + i √ 19 / 2- 5 | 5- 9 / 2 + i √ 19 / 2 | . This has rref 1- 9 / 10 + i √ 19 / 10 | | . Therefore, E 11 / 2+ i √ 19 / 2 = span 9 / 10- i √ 19 / 10 1 , so a basis for this eigenspace is 9 / 10- i √ 19 / 10 1 . For E 11 / 2- i √ 19 / 2 , (c) Find a rotation-scaling matrix B (i.e., a rotation matrix that has been multiplied by a scalar) such that A is similar to B ....
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This note was uploaded on 11/22/2011 for the course MATH 255 taught by Professor Staff during the Spring '10 term at UMass (Amherst).

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math 235 hw8 - Homework 8, Due May 4 Adam Gamzon Related...

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