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  "body": "\n## Summary\n\nThe Loewner order is the [[partial order|0QV2JS9RJ3]] on [[Hermitian matrices|0P17NGQQMQ]] in which $A \\succeq B$ means that $A - B$ is [[positive semidefinite|01MMVW8CKC]].\nIt extends the usual order of real numbers to matrices, but it is not a total order.\nIt is the order used in semidefinite programming, matrix inequalities and the theory of operator monotone functions.\n\n## Explanations\n\n### L1 human\n\nFor numbers, $a \\ge b$ means that $a - b$ is not negative.\nFor Hermitian matrices, $A \\succeq B$ means that $A - B$ is not negative in the matrix sense: $x^* (A - B) x \\ge 0$ for every vector $x$.\nSo $A$ is bigger than $B$ in every direction.\nTwo matrices can be incomparable: $\\operatorname{diag}(1, 0)$ is bigger in the first direction and $\\operatorname{diag}(0, 1)$ in the second.\n",
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  "name": "Loewner order",
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