Given an arbitrary $m\times n$ matrix $A$ with elements $a_{ij}$ and $m\neq n$. Is there a way to scale the rows and columns of that given matrix such that $\sum_{i} a_{ij} = \vec{b} \in \mathbb{R}^{n}$ and $\sum_{j} a_{ij} = \mathbf{1} \in \mathbb{R}^{m}$ for a given $\vec{b}$?
I found this article but it deals only with symmetric matrices. I am glad about every help.
I already found a solution to my problem! This article states that the iteratively alternately scaling of rows and columns converges to a matrix $D_{1} A D_{2}$ with the wished proportions. This is the case if $a_{ij} > 0$.