Denote the generalized eigenvectors and define is the

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Unformatted text preview: spaces is invariant under the action of is the direct sum of the generalized eigenspaces corresponding to eigenvalues with positive 11 Chapter 2 part B real part and it is also invariant under the action of . Similarly, the direct sum . and are invariant. is We can consider the action of in each subspace by considering restricted operators. denotes the restriction of to , etc. This corresponds to the fact that is block diagonal. For example, can always be brought to Jordan canonical form. A system is linearly stable if all its solutions are bounded as always bounded. Lemma 2.9. If and is an such that matrix and . If then is , the stable space of , then there are constants . Consequently, Remark. This result is very reasonable. From [1], each component of the solution will be proportional to for some eigenvalue , and by hypothesis is chosen so that for each such eigenvalue . The maximum power of that appears in any component of is , where is the maximum multiplicity of any eigenvalue in . For sufficiently large, the exponentially decaying terms must dominate the powers...
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