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Unformatted text preview: d dt x y = 1 2 2 1 x y and sketch the phase portrait. 6. Find the general solution of d dt x y = 1 22 1 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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This note was uploaded on 02/23/2010 for the course CDS 140A taught by Professor Marsden during the Fall '09 term at Caltech.
 Fall '09
 Marsden

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