$$\mathbf{P}_{\sigma}\mathbf{A}\pmb{v}=\mathbf{P}_{\sigma}\mathbf{A}\sum_{i=1}^na_i\pmb{x}_i
=\sum_{i=1}^na_i\mathbf{P}_{\sigma}\mathbf{A}\pmb{x}_i=
\sum_{i=1}^na_i\mathbf{A}\mathbf{P}_{\sigma}\pmb{x}_i
=\mathbf{A}\mathbf{P}_{\sigma}\sum_{i=1}^na_i\pmb{x}_i
=\mathbf{A}\mathbf{P}_{\sigma}\pmb{v}.$$ As a result, $\mathbf{P}_{\sigma}\mathbf{A}=\mathbf{A}\mathbf{P}_{\sigma}$, and thus the permutation $\sigma$ is an automorphism of $G$.

In accordance with Lemma \[Lem-AutomorphismAndOperator\], we can describe automorphisms of $G$ and so the group ${\mathrm{Aut} \hspace{0.4mm} G}$ in terms of eigenspaces of $\mathbf{A}(G)$. Let $U$ be a non-trivial subspace in $\operatorname{\mathbb{R}}^n$. Set $${\mathrm{Aut} \hspace{0.4mm} U} = \{ \sigma \in S_n : \sigma U = U \}.$$ Then $$\label{Equ-AutG-EigSpace}
{\mathrm{Aut} \hspace{0.4mm} G} = \bigcap_{\lambda \hspace{0.5mm} \in  \hspace{0.5mm} {\mathrm{spec} \hspace{0.4mm} {\mathbf{A}( G )}}} {\mathrm{Aut} \hspace{0.4mm} V_{\lambda}}.$$ For convenience, we denote the right hand side of the equation above by ${\mathrm{Aut} \hspace{0.4mm} \oplus V_{\lambda}}$. The relation (\[Equ-AutG-EigSpace\]) shows us that each eigenspace uncovers some information useful in characterizing the ${\mathrm{Aut} \hspace{0.4mm} G}$ action on $[n]$. As we have seen, the family of partitions $\{ \Pi^*_v : v \in [n] \}$ is critical in determining the partition $\Pi_G^*$, so let us see how to gather information about the partition $\Pi^*_v(\lambda)$ of $[n]$, which is composed of orbits of $({\mathrm{Aut} \hspace{0.4mm} V_{\lambda}})_v$, where $\lambda \in {\mathrm{spec} \hspace{0.4mm} {\mathbf{A}( G )}}$.

Recall that a linear operator $\mathcal{T}$ on $\operatorname{\mathbb{R}}^n$ is said to be an [*isometry*]{} if $\| \mathcal{T} \pmb{v} \| = \| \pmb{v} \|$ for any 