# there are no non-square doubly stochastic matrices

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

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

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

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

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

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

Title | there are no non-square doubly stochastic matrices |
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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 |