The system has an unstable eigendirection along

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Unformatted text preview: zero such that x x x . Then 0. Now let z x x x . Then clearly, z is a solution of x P x , with z . Furthermore, y 0 is also a solution, with y . By the uniqueness theorem, z y 0 Hence x x on the entire interval 9 . Let y 0 . Going in the other direction is trivial. be any solution of x z x y P x x . It follows that x is also a solution. Now let x y . Then the collection of vectors x x x y constitutes vectors, each with components. Based on the assertion in Prob. Section , these vectors are necessarily linearly dependent. That is, there are constants not all zero such that x x x y x y , 0. From Prob. , we have x x 0 for all Now , otherwise that would contradict the fact that the first vectors are linearly independent. Hence y x x x , and the assertion is true. . Consider z have x z x x x x , and suppose that we also x Based on the assumption, ________________________________________________________________________ page 364 —————————————————————————— CHAPTER 7. —— x x x 0. The collection of vectors x is linearly independe...
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