2011sol2 - M346 Second Midterm Exam Solutions April 7 2011...

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Unformatted text preview: M346 Second Midterm Exam Solutions, April 7, 2011 1) The matrix A = parenleftbigg 1 4 βˆ’ 3 parenrightbigg has eigenvalues Ξ» 1 = 1 and Ξ» 2 = βˆ’ 4, with eigenvectors b 1 = parenleftbigg 1 1 parenrightbigg and b 2 = parenleftbigg 1 βˆ’ 4 parenrightbigg . Suppose that x ( n ) satisfies the system of equations x ( n + 1) = A x ( n ) for all n β‰₯ 0. a) If x (0) is β€œrandom” (meaning any nonzero vector that isn’t an eigenvector of A ), compute the limits lim n β†’βˆž x 1 ( n ) x 2 ( n ) and lim n β†’βˆž x 1 ( n +1) x 1 ( n ) . In other words, what is the asymptotic direction of x ( n ) and the asymptotic growth rate? Since |βˆ’ 4 | > | 1 | , the dominant eigenvalue is Ξ» 2 , with dominant eigenvec- tor b 2 . Asymptotically, x will point in the b 2 direction and grow by a factor of βˆ’ 4 each turn, so the two answers are βˆ’ 1 / 4 and βˆ’ 4, respectively. b) Now suppose that x (0) = parenleftbigg 5 βˆ’ 5 parenrightbigg . Find x ( n ) exactly for all n . Since x (0) = 3 b 1 +2 b 2 (which you can get from change-of-basis matrices, or from row reduction), x ( n ) = 3(1) n b 1 + 2( βˆ’ 4) n b 2 = parenleftbigg 3 + 2( βˆ’ 4) n 3 βˆ’ 8( βˆ’ 4) n parenrightbigg . 2) Let A = parenleftbigg 1 4 βˆ’ 3 parenrightbigg , exactly as in problem 1. Suppose that x ( t ) satisfies the differential equation d x /dt = A x ....
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This note was uploaded on 04/10/2011 for the course M 346 taught by Professor Radin during the Spring '08 term at University of Texas.

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2011sol2 - M346 Second Midterm Exam Solutions April 7 2011...

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