Let $E_\lambda$ an eigenspace of a matrix associated to the eigenvalue $\lambda$. Suppose that $g_iE_{\lambda}=E_{\lambda}$ for any element $g_i$ of a finite group $G$. Is it true that the linear span of any eigenvector orbit in $E_{\lambda}$ is $E_{\lambda}$ itself? i.e.: $$ span(g_i w,\;\;g_i\in G,w\in E_{\lambda})=E_{\lambda} $$
for any choice of an eigenvector $w\in E_{\lambda}$?
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