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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 2-2 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 T-1 k 1 k T k 8. Let T be an n n matrix satisfying k I-T k < 1. Show that T is invertible and that the series X k =0 ( I-T ) k converges to T-1 . 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 x-y y = x-2 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