# there are no non-square doubly stochastic matrices

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

 $\displaystyle\sum_{j=1}^{m}a_{ij}$ $\displaystyle=$ $\displaystyle 1,\quad i=1,\ldots,n,$ (1) $\displaystyle\sum_{i=1}^{n}a_{ij}$ $\displaystyle=$ $\displaystyle 1,\quad j=1,\ldots,m.$ (2)

Then $n=m$.

This is seen by summing equation (1) over $i=1,\ldots,n$ and equation (2) over $j=1,\ldots,m$. Then

 $\displaystyle\sum_{i=1}^{n}\sum_{j=1}^{m}a_{ij}$ $\displaystyle=$ $\displaystyle n,$ $\displaystyle\sum_{i=1}^{n}\sum_{j=1}^{m}a_{ij}$ $\displaystyle=$ $\displaystyle m,$

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

Title there are no non-square doubly stochastic matrices ThereAreNoNonsquareDoublyStochasticMatrices 2013-03-22 15:11:00 2013-03-22 15:11:00 matte (1858) matte (1858) 8 matte (1858) Result msc 15A51 msc 60G99