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Unformatted text preview: d dt ³ x y ´ = ³ 1 2 2 1 ´³ x y ´ and sketch the phase portrait. 6. Find the general solution of d dt ³ x y ´ = ³ 1 22 1 ´³ x y ´ and sketch the phase portrait. 7. Let T be an invertible n × n matrix and k T k be its operator norm. Show that k T1 k ≥ 1 k T k 8. Let T be an n × n matrix satisfying k IT k < 1. Show that T is invertible and that the series ∞ X k =0 ( IT ) k converges to T1 . 2 9. Write the matrix A = 2 0 0 1 2 0 0 1 2 as the sum of two commuting matrices and use this to compute e A . 10. Show that all solutions of the linear system ˙ x =2 xy ˙ y = x2 y ˙ z =z converge to the origin as t → ∞ ....
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 Fall '09
 Marsden
 Equations, Conservation Of Energy, Summation, phase portrait, Dynamical systems, Laurence Fishburne

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