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  "body": "\n## Summary\n\nMajorization compares two vectors of real numbers with the same sum by how spread out their entries are.\nA vector $x$ majorizes $y$ when, for each $k$, the sum of the $k$ largest entries of $x$ is at least that of $y$.\nEquivalently, $y$ is obtained from $x$ by a [[doubly stochastic matrix|JJA90EB2PS]], that is, by averaging.\nIt organises many inequalities, including the [[Schur–Horn theorem|Y553MSYDAH]] on diagonals and eigenvalues of Hermitian matrices.\n\n## Explanations\n\n### L1 human\n\nThink of a fixed amount of money shared among $n$ people.\nOne sharing majorizes another when it is less equal: the richest person, the two richest people, and so on, always hold at least as much.\nMoving an amount from a richer person to a poorer one, no larger than the gap between them, gives a sharing that is majorized by the old one.\nFor example, $(1, 0, 0)$ majorizes $(\\frac{1}{2}, \\frac{1}{2}, 0)$, which majorizes $(\\frac{1}{3}, \\frac{1}{3}, \\frac{1}{3})$.\n",
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