quiz2_S06_solutions

quiz2_S06_solutions - Solutions to Quiz 2 1. State of...

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Unformatted text preview: Solutions to Quiz 2 1. State of system at time t ∈ N : ~x ( t ) = A t ~x (0) = 2 λ t 1 ~v 1 . In the first timestep, the vector ~x (0) is projected onto the line L 1 parallel to ~v 1 . After this step, if λ 1 > 1 resp. 0 < λ 1 < 1 the state proceedes expanding resp. shrinking along L 1 . If λ 1 = 1, it is stationary after the first step. 2. Characteristic polynomial det ( A- λ 1 ) = det 5- λ 1- 5 2 1- λ 8 2- 7- λ =- 5 det 2 1- λ 8 2- (7 + λ ) det 5- λ 1 2 1- λ =- 5 (4- 8- 8 λ )- (7 + λ ) ( 5- 6 λ + λ 2- 2 ) =- λ 3- λ 2- λ- 1 One solution of det ( A- λ 1 ) = 0 is λ =- 1. Factorization hence yields- λ 3- λ 2- λ- 1 =- ( λ + 1)( λ 2 + 1) Hence λ =- 1 is only real eigenvalue. 3. The kernel of A is ker 1 1 3 2 1 2 1 = ker 1 1 3 2 1 1 1 = ker 1 2 1 1 1 1 = span - 1- 1 1 | {z } ~v 1 , - 2- 1 1 | {z } ~v 2...
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This note was uploaded on 04/28/2008 for the course MAT 202 taught by Professor Staff during the Spring '08 term at Princeton.

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quiz2_S06_solutions - Solutions to Quiz 2 1. State of...

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