& 1/2\end{array}\right) ;\quad \det \mathbb{M} = 1/100\ , \label{A133} 
$$whose eigenvectors and eigenvalues are $$\left( 
\begin{array}{c}
1 \\ 
1 \\ 
1\end{array}\right) \leftrightarrow \chi_{PF} = 1,\quad \left( \begin{array}{r}
-2 \\ 
1 \\ 
1\end{array}\right) \leftrightarrow \chi_{2}=\frac{1}{10},\  \text{multiplicity 2}\ ,
\label{A14}$$ $\chi_{PF}$ is the Perron-Frobenius eigenvalue and $\chi_{2}$ is degenerate, with only one linearly independent eigenvector. The decomposition of matrix (\[A133\]) in the Jordan form is $$\mathbb{M} =\mathbb{RFR}^{-1}=\left( 
\begin{array}{rrr}
1 & 0 & 1 \\ 
1 & 5/2 & -1/2 \\ 
1 & -5/2 & -1/2\end{array}\right) \left( 
\begin{array}{rrr}
1 & 0 & 0 \\ 
0 & \chi_2 & 0 \\ 
0 & 1 & \chi_2\end{array}\right) \left( 
\begin{array}{rrr}
1/3 & 1/3 & 1/3 \\ 
0 & 1/5 & -1/5 \\ 
2/3 & -1/3 & -1/3\end{array}\right) 
\label{A15}$$ where the matrix $\mathbb{F}$ is not diagonal although the eigenvalues remain in the main diagonal. We can rewrite the matrix (\[A15\]) as $$\mathbb{M} = \mathbb{RDR}^{-1}+f\left( 1\right) \mathbb{L}
= \mathbb{C}_{1}\allowbreak +\chi _{2}\mathbb{C}_{2} +\chi
_{2}\mathbb{C}_{3}
\label{A16}$$ where the matrices $\mathbb{C}_{i}$ are $$\mathbb{C}_{1}=\frac{1}{3}\left( 
\begin{array}{ccc}
1 & 1 & 1 \\ 
1 & 1 & 1 \\ 
1 & 1 & 1\end{array}\right) ,\quad \mathbb{C}_{2}=\left( 
\begin{array}{rrr}
2/3 & -1/3 & -1/3 \\ 
-1/3 & 2/3 & -1/3 \\ 
-1/3 & -1/3 & 2/3\end{array}\right) 