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Unformatted text preview: M346 Second Midterm Exam, April 7, 2011 1) The matrix A = parenleftbigg 1 4 3 parenrightbigg has eigenvalues 1 = 1 and 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 isnt 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 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?...
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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 at Austin.
 Spring '08
 RAdin
 Linear Algebra, Algebra, Eigenvectors, Equations, Vectors

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