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  "body": "\n## Summary\n\nA doubly stochastic matrix is a square [[nonnegative matrix|JPQCCFCY8X]] in which every row and every column sums to $1$.\n[[Permutation matrices|08SP3DF8NB]] are doubly stochastic, and by the [[Birkhoff–von Neumann theorem|3KY1RMEDMJ]] every doubly stochastic matrix is a convex combination of them.\nThese matrices describe averaging processes and appear in [[majorization|5K7VT2VKZT]], assignment problems and Markov chains with a uniform stationary distribution.\n\n## Explanations\n\n### L1 human\n\nImagine $n$ glasses of water and a rule that pours fixed fractions of each glass into the others.\nSuppose that no water is lost, and that glasses which start equally full stay equally full.\nThen the table of fractions is doubly stochastic.\nEach row and each column of the table adds up to $1$.\nFor example, $\\frac{1}{2}\\begin{pmatrix} 1 \u0026 1 \\\\ 1 \u0026 1 \\end{pmatrix}$ mixes two glasses evenly.\n",
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