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# The relevant tra jec tory is the one crossing the p

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Unformatted text preview: aphs of x(t) = et/2 cos(3t/2) and y (t) = −(3/2)et/2 sin(3t/2) are “anti-damp ed” sinusoids, with increasing amplitude. The relevant tra jec­ tory is the one crossing the p ositive x �axis �half � way out. The values of u(t) are u − 23 = � � � � � �� −1 0 e−�/3 , u − 3 = e − � /6 , 0 3/2 �� � � �� 1 0 u(0) = , u � = e�/6 , 3 0 −3 / 2 � � � �� −1 u 23 = e�/3 . 0 � � 1b (b) [8] With A = , pA (�) = �2 − 2� + 1 = (� − 1)2 , so we have a rep eated eigenvalue 01 � � 0b �1 = 1. To ﬁnd an eigenvector form A − �1 I = . A nonzero eigenvector is given (for 00 �� 1 any b) by v = . If b = 0, the eigenvectors for value � 1 are exactly the multiples of v � 0 (the matrix is defective), but for b = 0, A = I and an� vector is an eigenvector (the matrix is �y c complete). Wh...
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## This note was uploaded on 01/22/2014 for the course MATH 18.03 taught by Professor Vogan during the Spring '09 term at MIT.

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