$N$ of the matrix $\mathbb{M}$. Notwithstanding, we can write the decomposition in a form similar to (\[A8\]), $\mathbb{M}=\mathbb{RFR}^{-1}$, which is the so-called *Jordan canonical form*, where the matrix $\mathbb{R}$ is different from $\mathbb{Q}$, and the matrix $\mathbb{F}$, besides having the eigenvalues of $\mathbb{M}$ in the diagonal line, will contain, additionally, in the adjacent line parallel to the main diagonal, the number $1$ in the entries of the blocks containing the degenerate eigenvalues, while the other entries are filled with zeros. The theory is due to the french mathematician Camille Jordan. For a rigorous mathematical presentation of the matter we recommend the reader to consult, for instance, Ref. [@horn2013].

The decomposition of a stochastic matrix in the Jordan form can be cast as $$\mathbb{M}=\mathbb{RDR}^{-1} + \mathbb{L},\quad \text{and} \quad 
\mathbb{M}^{n}=\mathbb{R} \mathbb{D}^{n}\mathbb{R}^{-1} +f(n)\mathbb{L},  
\label{A12}$$ where $\mathbb{D}$ is still a diagonal matrix containing the eigenvalues, $\mathbb{L}=\mathbb{R}\mathbb{O}\mathbb{R}^{-1}$ with $\mathbb{O}$ the matrix containing $1$’s (and $0$’s) in a secondary diagonal line of matrix $\mathbb{F}$ and $f(n)$ is a function only of $n$. We illustrate this case through an example.

**** Let us consider the doubly stochastic matrix $$\begin{aligned}
\mathbb{M} &=&\left( 
\begin{array}{ccc}
2/5 & 1/2 & 1/10 \\
3/10 & 3/10 & 2/5 \\
3/10 & 1/5 