I have the following problems when solving a linear equation.
Let $A=(a_{i,j})_{n \times n}$ be a non-negative matrix with $a_{i,j} \in (0,1)$, and let $0 (I) The first component of $x$ is 1, that is $x_1=1$. (II) The other components of $x$ (except the first entry of $x$) satisfies the following equation: $$r \cdot Ax=x .$$ Or equivalently, both (I) and (II) tell that $x$ satisfies the following equation:
$$max \{r \cdot Ax,e_1\} =x $$ where $max$ is entry-wise maximum operator, and $e_1={(1,0,\cdots,0)}^T$ . Based on such a defintion of $x$, I want study the relations between $x$ and the vector $y$ that satisfies $r \cdot Ay=y$ (including the first entry of $y$).
In other words, can we compute $x$ from $y$ ? I would really appreciate any suggestions.