Timeline for How do I find non-linear sets that are invariant under a certain linear transformation?
Current License: CC BY-SA 3.0
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Aug 18, 2011 at 13:40 | history | edited | user17119 | CC BY-SA 3.0 |
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Aug 18, 2011 at 13:10 | comment | added | Donu Arapura | Initially, I thought you meant pointwise invariance $\forall s\in S, T(s)=s$ , but I guess you mean $T(S)\subseteq S$. You can disregard my previous comment. | |
Aug 18, 2011 at 12:58 | history | edited | user17119 | CC BY-SA 3.0 |
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Aug 18, 2011 at 12:46 | comment | added | user17119 | Donu, why would the linear span of any invariant set have to lie in V? Take the eigenspace spanned by the eigenvector corresponding to some other eigenvalue. It is invariant under T, but does not lie in V. | |
Aug 18, 2011 at 2:51 | comment | added | Donu Arapura | Or do you mean sets such that $T(S)\subset S$? | |
Aug 18, 2011 at 2:42 | comment | added | Donu Arapura | It's clear isn't it? Let $V$ be the $+1$-eigenspace (which might be $0$) of $T$. Then any subset of $V$ is invariant. Conversely, the linear span of an invariant set would have to lie in $V$. | |
Aug 18, 2011 at 2:29 | history | edited | Yemon Choi |
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Aug 18, 2011 at 2:28 | comment | added | Yemon Choi | But what kinds of set are you looking for? It seems like you want algebraic sets of some kind (e.g. solutions to systems of equations) and not just any old set. | |
Aug 18, 2011 at 1:49 | comment | added | user17119 | What I'm looking for is a reference to the mathematical area/technique that deal with such problems. | |
Aug 18, 2011 at 1:39 | comment | added | Yemon Choi | Without further conditions on $T$ or some maximality condition on your non-linear set $S$, this seems tricky to answer sensibly. (Case in point: take T to be the identity and S to be anything very like a whale.) Is there a more precise version of this question, specifying domains etc, that you coudl pose? | |
Aug 18, 2011 at 1:11 | history | asked | user17119 | CC BY-SA 3.0 |