For this problem I'm asking for references or any toolbox on python that can help me, since I'm absolutly lost, even how to search some help on internet.
Consider $P$ a $16\times 16$ matrix, wich is know except by $15$ parameters. Consider some row verctor $\mu$ that in my case is $\begin{pmatrix}1& 0 &0 &0 &... &0\end{pmatrix}$, and some target vector $d\in \mathbb{R}^{16}$. The goal is to determine the $15$ parameters such that $\mu P^6=d$
The context is that I have a Markov Chain $(M_i)_{i=0}^{6}$ with 16 states, where the transition matrix $P$ have some parameters, and I'd like to find those parameters in order to get some given distribution on time 6.
Thank you.
$\mu P^6$ is the first row of $P^6$, that is the probability distribution at time $6$ when we come from the state $1$ at time $0$. Necessarily $(*)$ "your $15$ unknown entries must explicitly appear in this first row"; else you cannot determine them! If $(*)$ is true, then you have to solve a system of $16-1=15$ equations (because the sum of any row is $1$) in $15$ unknowns. If you are lucky, then the degrees of your equations may be low ($1$ for example if you are very lucky).