My question concerns the following systems of linear equations: $x = 1 + Ax$ and $y = 1 + By$, where 1 denotes the $n$x1 vector of 1's and $A$ and $B$ are $n$x$n$ substochastic matrices that satisfy the following:
- $A_{i,j} \geq 0$ and $B_{i,j} \geq 0$ for each $i,j$
- $\sum\limits_{j=1}^n A_{i,j} \leq 1$ and $\sum\limits_{j=1}^n B_{i,j} \leq 1$ for each $i$.
- For each $i,j$: $\sum\limits_{k=1}^j A_{i,k} \geq \sum\limits_{k=1}^j B_{i,k}$.
Intuitively, it seems very clear to me that these conditions imply that $x_i \geq y_i$ for each $i = 1, ..., n$, but I'm not sure how to prove this. Perhaps there is some elementary linear algebra result that I am not aware of. Hope you can help me!