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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $A$ be an $n \\times n$ [[doubly stochastic matrix|JJA90EB2PS]].\nSo $A$ has real nonnegative entries, and its rows and columns each sum to $1$.\nThen $A$ is a convex combination of $n \\times n$ [[permutation matrices|0Q5SM60HTH]].\nThat is, there are permutation matrices $P_1, \\dots, P_k$ and real numbers $\\theta_1, \\dots, \\theta_k \\ge 0$ with $\\theta_1 + \\cdots + \\theta_k = 1$ such that\n$$A = \\theta_1 P_1 + \\cdots + \\theta_k P_k.$$\n\n## Remarks\n\nThe proof is by induction on the number of nonzero entries of $A$.\nForm the bipartite graph whose two sides are the rows and the columns, with an edge for each nonzero entry.\nAny $r$ rows carry total mass $r$, and this mass lies in the columns joined to them, which carry at most $1$ each.\nSo [[Hall's marriage theorem|08WCB90WND]] gives a perfect matching.\nIt is a permutation matrix $P$ supported on nonzero entries of $A$.\nIf $\\theta$ is the least entry of $A$ on this support and $A \\ne P$, then $(A - \\theta P)/(1 - \\theta)$ is doubly stochastic with fewer nonzero entries.\nFor example,\n$$\\begin{pmatrix} 1/2 \u0026 1/2 \\\\ 1/2 \u0026 1/2 \\end{pmatrix} = \\frac12 \\begin{pmatrix} 1 \u0026 0 \\\\ 0 \u0026 1 \\end{pmatrix} + \\frac12 \\begin{pmatrix} 0 \u0026 1 \\\\ 1 \u0026 0 \\end{pmatrix}.$$\nEquivalently, the set of doubly stochastic matrices, the Birkhoff polytope, is the [[convex hull|0TCHAS04WK]] of the $n!$ permutation matrices, and these are its vertices.\n\n## Explanations\n\n### L1 human\n\nA doubly stochastic matrix spreads one unit from each row over the columns so that every column also receives one unit.\nThe theorem says that every such spreading is a mixture of perfect one-to-one assignments.\nSo a fractional assignment can always be written as a weighted average of whole assignments.\n\n### L2 human\n\nThe theorem is the matrix form of Hall's and Kőnig's theorems on bipartite matchings.\nIt shows that the linear program for the assignment problem has a permutation matrix among its optimal solutions.\nIt is also used in the theory of majorization and in proofs of inequalities for permanents and eigenvalues.\n",
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