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

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.