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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $x, y \\in \\R^n$.\nWrite $x^{\\downarrow}$ for the vector with the entries of $x$ in decreasing order, so $x^{\\downarrow}_1 \\ge x^{\\downarrow}_2 \\ge \\cdots \\ge x^{\\downarrow}_n$.\nThe vector $x$ *majorizes* $y$, written $y \\prec x$, if:\n\n1. $\\sum_{i=1}^{k} y^{\\downarrow}_i \\le \\sum_{i=1}^{k} x^{\\downarrow}_i$ for each $k = 1, \\ldots, n - 1$.\n2. $\\sum_{i=1}^{n} y_i = \\sum_{i=1}^{n} x_i$.\n\n## Remarks\n\nExample: $y = (\\frac{1}{2}, \\frac{1}{2}, 0) \\prec x = (1, 0, 0)$.\nCheck: $\\frac{1}{2} \\le 1$ for $k = 1$, $1 \\le 1$ for $k = 2$, and both sums equal $1$.\nNon-example: $(1, 0) \\not\\prec (2, 0)$, because clause 1 holds but the sums $1$ and $2$ differ, so clause 2 fails.\nThe relation is a preorder: $y \\prec x$ and $x \\prec y$ hold together exactly when $y$ is a permutation of $x$.\nBy a theorem of Hardy, Littlewood and Pólya, $y \\prec x$ if and only if $y = Dx$ for some $n \\times n$ [[doubly stochastic matrix|JJA90EB2PS]] $D$.\n",
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