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Unformatted text preview: 1 CDS 140a: Homework Set 3 Due: Friday, October 23th, 2009. 1. Solve the system ˙ x = x y ˙ y = x + 3 y for given initial conditions ( x ,y ). 2. Do all solutions of the system ˙ x = x + y + z ˙ y = y + 2 z ˙ z = 2 z converge to the origin as t → ∞ ? 3. Do all solutions of the system ˙ x = x + y + z ˙ y = y + 2 z ˙ z = 2 z converge to the origin as t → ∞ ? 4. Find the Jordan canonical form, the S + N decomposition and the matrix exponential of the matrix 1 0 0 1 1 0 0 1 5. Find the generalized eigenspaces of the matrix in the preceding problem and show directly that these subspaces are invariant under the equation ˙ x = Ax and span all of R 3 . 6. Find the Jordan canonical form, the S + N decomposition and the matrix exponential of the matrix 1 1 0 0 1 0 0 0 0 1 0 0 1 7. Find the generalized eigenspaces of the matrix in the preceding problem and show directly that these subspaces are invariant under the equation ˙ x = Ax and span all of R 4 . 2...
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 Fall '09
 Marsden
 Canonical form, Jordan Canonical Form, Matrix exponential, −∞. ˙

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