This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
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?...
View Full Document