$\mathbf{A}(G / \Pi)$. To be precise, ${\mathrm{spec} \hspace{0.4mm}  {\mathbf{A}( G / \Pi )} } \subseteq {\mathrm{spec} \hspace{0.4mm}  {\mathbf{A}( G )} }$, and if $\pmb{x}_{\lambda}$ is an eigenvector of $\mathbf{A}(G / \Pi)$, corresponding to the eigenvalue $\lambda$, then $\mathbf{R}_{\Pi} \pmb{x}_{\lambda}$ is an eigenvector of $\mathbf{A}(G)$, corresponding to $\lambda$ also, where $\mathbf{R}_{\Pi}$ is the characteristic matrix of $\Pi$. Accordingly, we say that the eigenvector $\pmb{x}_{\lambda}$ of $\mathbf{A}(G / \Pi)$ [*“lifts”*]{} to the eigenvector $\mathbf{R}_{\Pi} \pmb{x}_{\lambda}$ of $\mathbf{A}(G)$. Moreover all eigenvectors of $\mathbf{A}(G)$ could be divided into two classes: those that are constant on every cell of $\Pi$ and those that sum to zero on each cell of $\Pi$. As one can readily see, the first class consists of vectors lifted from eigenvectors of $\mathbf{A}(G / \Pi)$. In other words, if $\Pi = \{ C_1,\ldots,C_t \}$ is an equitable partition and $x$ and $y$ are two vertices of $G$ belonging to the same cell of $\Pi$, then $$\label{Def-EquitablePartition}
\langle \pmb{e}_x,{\mathrm{proj}\big[ V_\lambda \big]}(\pmb{R}_j) \rangle = \langle \pmb{e}_y,{\mathrm{proj}\big[ V_\lambda \big]}(\pmb{R}_j) \rangle, ~\forall~\lambda\in\mathrm{spec}~\mathbf{A}(G) \mbox{\it ~and } j\in [t],$$ where $\pmb{R}_j$ is the characteristic vector of $C_j$ and the vector ${\mathrm{proj}\big[ V_\lambda \big]}(\pmb{R}_j)$ is the orthogonal projection of $\pmb{R}_j$ onto the eigenspace $V_{\lambda}$. As we shall see 