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Can eigenvectors and eigenvalues be seen as limits in the same sense as fixed points?

What I mean by the same sense as fixed points is that if $G$ is group we can consider limits to ${\bf Set}^G$, or $G$-Sets. If $S$ is the $G$-Set the limits is the equalizer of: $S\rightrightarrows \prod_{g\in G}S_g$ where one arrow sends $s\in S$ to $(g(s),\ldots)\in \prod_{g\in G}S_g$ and the other sends $s$ to $(s,\ldots)$

The equalizer consists points in $S$ that are unchanged by the action of $G$. I'm wondering if there is any sensible way to interpret the eigenvectors and eigenvalues as limits in a similar fashion.

Thanks in advance

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