below, the relation above is also sufficient for being equitable.

\[Lemma-EquitablePartProj\] Let $\Pi = \{ C_1,\ldots,C_t \}$ be a partition of $V(G)$. Then $\Pi$ is equitable if and only if for any two vertices $x$ and $y$ belonging to the same cell of $\Pi$, the relation (\[Def-EquitablePartition\]) holds.

We have discussed the necessity of our assertion, so let us show the sufficiency now. Obviously, the vectors $\pmb{R}_1,\ldots,\pmb{R}_t$ comprise an orthogonal basis of $U_\Pi$, which is the column space of $\mathbf{R}_{\Pi}$. To prove $U_\Pi$ is $\mathbf{A}(G)$-invariant, it suffices to show that $\mathbf{A}(G) \pmb{R}_k$ ($1\leq k \leq t$) can be written as a linear combination of $\pmb{R}_1,\ldots,\pmb{R}_t$.

In fact, $$\begin{aligned}
\mathbf{A}(G) \pmb{R}_k 
& = \mathbf{A}(G) \left( \sum_{\lambda \hspace{0.2mm} \in \hspace{0.2mm} \mathrm{spec} \hspace{0.2mm} \mathbf{A}(G)} {\mathrm{proj}\big[ V_{\lambda} \big]}( \pmb{R}_k ) \right) \\
& = \sum_{\lambda} \mathbf{A}(G){\mathrm{proj}\big[ V_{\lambda} \big]}( \pmb{R}_k ) \\
& = \sum_{\lambda} \lambda\cdot{\mathrm{proj}\big[ V_{\lambda} \big]}( \pmb{R}_k ).\end{aligned}$$ In accordance with our assumption, one can readily see that ${\mathrm{proj}\big[ V_{\lambda} \big]}( \pmb{R}_k )$ can be expressed as a linear combination of $\pmb{R}_1,\ldots,\pmb{R}_t$, so is $\mathbf{A}(G) \pmb{R}_k$.

Clearly Lemma \[Lemma-EquitablePartProj\] shows us that if $\Pi$ is an equitable partition and $C$ is a cell of $\Pi$, then the projection ${\mathrm{proj}\big[ V_{\lambda} \big]}( \pmb{R}_C )$ 