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2011test2web

# 2011test2web - M346 Second Midterm Exam April 7 2011 01 1...

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M346 Second Midterm Exam, April 7, 2011 1) The matrix A = parenleftbigg 0 1 4 - 3 parenrightbigg has eigenvalues λ 1 = 1 and λ 2 = - 4, with eigenvectors b 1 = parenleftbigg 1 1 parenrightbigg and b 2 = parenleftbigg 1 - 4 parenrightbigg . Suppose that x ( n ) satisfies the system of equations x ( n + 1) = A x ( n ) for all n 0. a) If x (0) is “random” (meaning any nonzero vector that isn’t an eigenvector of A ), compute the limits lim n →∞ x 1 ( n ) x 2 ( n ) and lim n →∞ x 1 ( n +1) x 1 ( n ) . In other words, what is the asymptotic direction of x ( n ) and the asymptotic growth rate? b) Now suppose that x (0) = parenleftbigg 5 - 5 parenrightbigg . Find x ( n ) exactly for all n . 2) Let A = parenleftbigg 0 1 4 - 3 parenrightbigg , exactly as in problem 1. Suppose that x ( t ) satisfies the differential equation d x /dt = A x . a) How many stable and how many unstable modes does this system have? What is the dominant eigenvalue, and what is the dominant eigenvector? For typical initial conditions, compute lim t →∞ x 1 ( t ) /x 2 ( t ).

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