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2009btest2web - M346 Second Midterm Exam 1 Diagonalization...

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Unformatted text preview: M346 Second Midterm Exam, October 23, 2009 1) Diagonalization: a)(10 pts) Find the eigenvalues and eigenvectors of A = parenleftbigg 3 8 2- 3 parenrightbigg b) (10 pts) Compute e Bt , where B = parenleftbigg 1 2 2 4 parenrightbigg . c) (10 pts) Find the eigenvalues of C = 2 2 3 7 2 5 2 8 6 16 4- 6 . You do not need to find the eigenvectors. 2. Consider the system of equations x ( n +1) = A x ( n ), where A = parenleftbigg 1 2- 1 parenrightbigg , with initial condition x (0) = parenleftbigg- 1 5 parenrightbigg . a) (15 pts) Find x ( n ) for all n . Be as explicit as possible. b) (5 pts) Find lim n →∞ x 1 ( n ) x 2 ( n ) . c) (5 pts) Find lim n →∞ x 1 ( n +1) x 1 ( n ) . 3. a) (15 pts) Consider the system of nonlinear coupled differential equations dx 1 dt = x 1 (3- 2 x 1- x 2 ) dx 2 dt = x 2 (3- x 1- 2 x 2 ) . This system of equations has four fixed points, namely (0 , 0) T , (3 / 2 , 0) T , (0 , 3 / 2) T , and (1 , 1) T . These equations describe competition between two....
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2009btest2web - M346 Second Midterm Exam 1 Diagonalization...

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