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
diagonal. For example, can always be brought to Jordan canonical form. A system is linearly stable if all its solutions are bounded as
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 , each component of the solution will be
for some eigenvalue , and by hypothesis
is chosen so that
for each such eigenvalue
. The maximum power of that appears in any component
, where is the maximum multiplicity of any eigenvalue in
sufficiently large, the exponentially decaying terms must dominate the powers...
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- Fall '14