\mathbb{X}^{\left( 3\right) } \end{array}\right)$, $$\mathbb{Q}=\left( 
\begin{array}{rrr}
{0.577350} & {0.576331} & {0.660268} \\ 
{0.577350} & -{0.802915} & {0.039495} \\ 
{0.577350} & -{0.152215} & -{0.749991}\end{array}\right) ,  \label{C4}$$ and $\mathbb{D}=\mathrm{Diag}[1.0,0.367945,-0.067945]$. The matrix $\mathbb{M}^{n}=\mathbb{QD}^{n}\mathbb{Q}^{-1}=\left( 
\begin{array}{ccc}
\mathbb{Y}^{\left( 1,n\right) } & \mathbb{Y}^{\left( 2,n\right) } & \mathbb{Y}^{\left( 3,n\right) }\end{array}\right)$ so that the three columns are $$\mathbb{Y}^{\left( 1,n\right) }=\underbrace{\frac{{\ 11}}{{\ 27}}\left( 
\begin{array}{c}
{\ 1} \\ 
{\ 1} \\ 
{\ 1}\end{array}\right) }_{\text{stationary mode}}+\underbrace{\left( 
\begin{array}{r}
{0.304793} \\ 
{-0.424622} \\ 
{-0.326908}\end{array}\right) \left( {\lambda _{2}}\right) ^{n}+\left( 
\begin{array}{r}
{0.287800} \\ 
{0.017215} \\ 
{-0.326908}\end{array}\right) \left( {\ \lambda _{3}}\right) ^{n}}_{\text{ decaying modes}}
\label{C11}$$ $$\mathbb{Y}^{\left( 2,n\right) }=\underbrace{\frac{{\ 2}}{{\ 9}}\left( 
\begin{array}{c}
1 \\ 
1 \\ 
1\end{array}\right) }_{\text{stationary mode}}+\underbrace{\left( 
\begin{array}{r}
{-0.544452} \\ 
{0.758503} \\ 
{0.143795}\end{array}\right) \left( {\ \lambda _{2}}\right) ^{n}+\left( 
\begin{array}{r}
{0.322230} \\ 
{0.019275} \\ 
{-0.366017}\end{array}\right) \left( {\ \lambda _{3}}\right) ^{n}}_{\text{decaying modes}}
\label{C12}$$ $$\mathbb{Y}^{\left( 3,n\right) }=\underbrace{\frac{{\ 10}}{{\ 27}}\left( 
\begin{array}{c}
1 \\ 
1 \\ 
1\end{array}\right) }_{\text{stationary mode}}+\underbrace{\left( 
\begin{array}{r}
{0.239659} \\ 
{-0.333880} \\ 
{-0.063296}\end{array}\right) \left( {\ \lambda _{2}}\right) ^{n}+\left( 
\begin{array}{r}
{-0.610029} \\ 
{-0.036490} \\ 
{0.692926}\end{array}\right) \left( {\ \lambda _{3}}\right) ^{n}}_{\text{decaying modes}}\ .
\label{C13}$$ We further write the spectral decomposition of matrix (\[C1\]) in terms of the modes or regimes $\mathbb{M}^{n}\mathbb{=C}_{1}+\mathbb{C}_{2}\left( {\
\lambda _{2}}\right) ^{n}+\mathbb{C}_{3}\left( {\ \lambda _{3}}\right) ^{n}$, where the three matrices and their determinants are $$\mathbb{C}_{1}=\left( 
\begin{array}{ccc}
11/27 & 