there are no non-square doubly stochastic matrices

Suppose $A=(a_{ij})$ is a $n\times m$ matrix with nonnegative entries suchthat

${\displaystyle \sum _{j=1}^m}{a}_{ij}$$=$$1,i=1,\mathrm{\dots },n,$(1)

${\displaystyle \sum _{i=1}^n}{a}_{ij}$$=$$1,j=1,\mathrm{\dots },m.$(2)

Then $n=m$.

This is seen by summing equation(1) over $i=1,\mathrm{\dots },n$ andequation (2) over $j=1,\mathrm{\dots },m$. Then

${\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^m}{a}_{ij}$$=$$n,$

${\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^m}{a}_{ij}$$=$$m,$

and since the right hand sides coincide, it follows that $n=m$.