Consider the following equation
$A x = \log (Px)$,
where $x = [x_1, \dots, x_n]^T$ with $ 0 < x_i < 1$ and $\sum_{i} x_i = 1$. In addition, $A \in \mathbb{R}^{n \times n}$ and $P$ is a stochastic matrix with $\sum_{i=1}^n P_{ij} = 1$ for every $j \in \{1, \dots n\}.$
We want to solve the above equation for $A$ given the stochastic matrix $P$ and for any vector $x$. Is this possible?
Otherwise, can we find an $A$ that minimizes the following optimization problem?
$\min\limits_{A} \|Ax - \log(Px)\|_2$
for any $\quad 0 < x_i < 1$ for $x_i \in \{1, \dots, n\},$ $\sum_{i=1}^n x_i = 1$
I'd appreciate any pointers.