A matrix denoted by $(a_{ij})_{m \times n}$ is said to be doubly stochastic if:

$$ \sum_{i}{a_{ij}} = \sum_{j}{a_{ij}} = 1 $$

I am trying to prove that such a matrix is a square matrix. I thought of multiplying or adding 2 of such matrices together but this leads to nowhere. I also wrote down down a generalised doubly stochastic matrix in this form:

$$ \begin{bmatrix}a_{11} & a_{12} & \cdots & a_{1n}\\a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} $$

I can't see any relationship between the different rows and columns from this form. Could anyone please give me some hints?


Since the sum of rows is $1$ and there are $n$ rows then the sum of all entries is $n$ similarly the sum of column entries is also $1$ and suppose that there are $m$ columns then the sum of matrix entries is $m$ but the matrix sum is invariant. Hence $m=n$


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