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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $A = (a_{ij})$ be a real $n \\times n$ [[matrix|0JXZKX526R]].\nThe matrix $A$ is *doubly stochastic* if:\n\n1. $a_{ij} \\ge 0$ for all $i$ and $j$.\n2. $\\sum_{j=1}^{n} a_{ij} = 1$ for each row index $i$.\n3. $\\sum_{i=1}^{n} a_{ij} = 1$ for each column index $j$.\n\n## Remarks\n\nClause 1 says that $A$ is a [[nonnegative matrix|JPQCCFCY8X]].\nWith $e = (1, \\ldots, 1)^T$, clauses 2 and 3 say $Ae = e$ and $e^T A = e^T$.\nExample: $A = \\frac{1}{2}\\begin{pmatrix} 1 \u0026 1 \\\\ 1 \u0026 1 \\end{pmatrix}$; all entries are $\\frac{1}{2} \\ge 0$, and each row and each column sums to $1$.\nEvery [[permutation matrix|08SP3DF8NB]] is doubly stochastic.\nNon-example: $\\begin{pmatrix} 1 \u0026 0 \\\\ 1 \u0026 0 \\end{pmatrix}$ satisfies clauses 1 and 2, but its columns sum to $2$ and $0$, so clause 3 fails.\nThe set of doubly stochastic $n \\times n$ matrices is the Birkhoff polytope.\n",
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