How to write spectral form of probability matrix I am trying understand Markov chain in genetics process. In book that I am using (Mathematical Population Genetics) (pag 87):  (P is matrix transition probability). $E_0$ and $E_M$ are absorbing states.
$P^{(t)}= P^t$
So, they said $P^t$ can be write in spectral form
$$P^t=\lambda^t_0 r_0 l'_0+ \lambda^t_1 r_1 l'_1 + \ldots \lambda^t_M r_M l'_M$$
where $\lambda$ are eigenvalues, $r$ and $l$ are right and left eigenvectors. Normalized so that  
$$ l'_i \textbf{r}_i= \sum_{j=0}^{M} l_{ij}r_{ij}=1$$
I do not understand how to write this spectral form. And the normalization step. 
 A: The following answer is based upon material found at the start of Chapter 1 of Ledermann and Reuter$^1$, a paper on JSTOR. The method shown is a method for matrices without multiple eigenvalues. It applies to any matrix with a full set of right and left eigenvectors ( no multiple eigenvalues).
Take our matrix $\mathbf{P}^t$ to be an $(M+1) \times (M+1)$ matrix with distinct eigenvalues. Let the $(M+1)$ row vectors $\mathbf{l}_i$ satisfy
\begin{equation}
\mathbf{l}_i\mathbf{P}^t=\lambda_i\mathbf{l}_I\tag{1}
\end{equation}
and the $(M+1)$ column vectors $\mathbf{r}_i^\prime$ satisfy
\begin{equation}
\mathbf{P}^t\mathbf{r}_i^\prime=\lambda_i\mathbf{r}_i^\prime\tag{2}
\end{equation}
,please note a prime is used to indicate a column vector and there is no $t$ superscript on any $\lambda$.
Combine the row vectors into a matrix $\mathbf{L}$
$$\mathbf{L}=\begin{bmatrix} \mathbf{l}_0 \\     \mathbf{l}_1 \\   \vdots \\  \mathbf{l}_M \\    \end{bmatrix}$$
and combine the column vectors into a matrix $\mathbf{R}$
$$\mathbf{R}=\begin{bmatrix} \mathbf{r}_0^\prime     \mathbf{r}_1^\prime   \cdots  \mathbf{r}_M^\prime \\    \end{bmatrix}$$
The column vectors $\mathbf{r_i^\prime}$ can be chosen such that
\begin{equation}\mathbf{L} \mathbf{R}=\mathbf{1}
\end{equation}
Apparently , this is possible because of the orthogonality properties of eigenvectors belonging to different eigenvalues and a multiple of an eigenvector is still an eigenvector belonging to the same eigenvalue.
So, we have in particular, that the column vectors can be chosen such that
\begin{equation}\mathbf{l}_i \mathbf{r}_i^\prime=1, i=0,1,\cdots,M \tag{3}
\end{equation}
Equations (1) and (2) can be combined  ( an explanation of this could be put in  'Other Details' ) using $\mathbf{L}$ and $\mathbf{R}$ into
\begin{equation}\mathbf{L} \mathbf{P}^t \mathbf{R}=\mathbf{\lambda}
\end{equation}
, where $\mathbf{\lambda}$ is a diagonal matrix with eigenvalues on the diagonal.
$\mathbf{ \lambda}=\begin{bmatrix}
\lambda_0 & 0 & \cdots & 0 \\
0 & \lambda_1 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & \lambda_M
\end{bmatrix}$
So we have,
\begin{align}
\mathbf{L} \mathbf{P}^t \mathbf{R}&=\mathbf{\lambda} \nonumber \\
\mathbf{L}^{-1}    (\mathbf{L} \mathbf{P}^t \mathbf{R} )  \mathbf{R}^{-1}         &=   \mathbf{L}^{-1}      \mathbf{\lambda}   \mathbf{R}^{-1}       \nonumber \\
 \mathbf{P}^t          &=   \mathbf{R}      \mathbf{\lambda}   \mathbf{L}      \nonumber 
\end{align}
where $\mathbf{L} \mathbf{R}=\mathbf{1}$ has been used.
The expression for $\mathbf{P}^{t}$ can be put in the form
\begin{equation} \mathbf{P}^t = \sum_{i=0}^{M} \mathbf{\lambda}_i   \mathbf{r}_i^\prime    \mathbf{l}_i \tag{4}
\end{equation}
this takes some effort to explain properly, pehaps this could be put in 'Other Details'? Please note $ \mathbf{r}_i^\prime    \mathbf{l}_i   $ is a $(M+1) \times (M+1)$ matrix.
Now changing some notation, putting a '$t$' superscript on any $\lambda$'s and using a prime to denote a row vector rather than a column vector, the results (4) and (3) may be expressed as
\begin{equation}
 \mathbf{P}^t = \mathbf{\lambda}_0^t   \mathbf{r}_0   \mathbf{l}_0^\prime+
 \mathbf{\lambda}_1^t   \mathbf{r}_1   \mathbf{l}_1^\prime+\cdots+
 \mathbf{\lambda}_M^t   \mathbf{r}_M  \mathbf{l}_M^\prime
\end{equation}
\begin{equation}   \mathbf{l}_i^\prime   \mathbf{r}_i =    \sum_{j=0}^{M}   [ l_i^{\prime}]_j [r_i]_j=1
\end{equation}
These equations are more consistent with how the question was asked, there is a problem with the question statement, in the use of the notation $l_{ij},r_{ij}   $. I use the notation $[ l_i^{\prime}]_j$ or $[r_i]_j$ to stand for the $j^{th}$ element of a row or column matrix respectively.
Reference:
1) On JSTOR
"https://www.jstor.org/stable/91569?seq=1#page_scan_tab_contents"
Accessed: 01-08-2019 10:23 UTC
Other Details
For further detail.
