there are no non-square doubly stochastic matrices

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

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

Title	there are no non-square doubly stochastic matrices
Canonical name	ThereAreNoNonsquareDoublyStochasticMatrices
Date of creation	2013-03-22 15:11:00
Last modified on	2013-03-22 15:11:00
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	8
Author	matte (1858)
Entry type	Result
Classification	msc 15A51
Classification	msc 60G99