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  "body": "\n## Summary\n\nA real function $f$ on an interval is operator monotone if it keeps the order between [[Hermitian matrices|0P17NGQQMQ]]: from $A \\le B$ it follows that $f(A) \\le f(B)$.\nHere $A \\le B$ is the [[Loewner order|9G9S92T1VJ]], and $f(A)$ is defined through the eigenvalues of $A$.\nEvery operator monotone function is a [[monotone function|0A7KV2WB7A]], but the converse fails, so operator monotonicity is a much stronger condition.\n[[Löwner's theorem|PA7ZQ6C6SP]] describes these functions through complex analysis.\n\n## Explanations\n\n### L1 human\n\nFor numbers, an increasing function keeps the order: if $a \\le b$ then $f(a) \\le f(b)$.\nHermitian matrices can also be compared: $A \\le B$ means that $B - A$ has no negative eigenvalues.\nAn operator monotone function is one that keeps this matrix order, for matrices of every size.\nThe square root $\\sqrt{t}$ and the logarithm $\\log t$ on $(0, \\infty)$ do this, while the square $t^2$ does not, even though it is increasing on $[0, \\infty)$.\n",
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