How to write spectral form of probability matrix

Question

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.