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  "body": "\n## Summary\n\nA Hermitian kernel is a complex-valued [[kernel|GSMBXT7FPK]] with $K(x, s) = \\overline{K(s, x)}$ for all $x$ and $s$.\nIt is the continuous analogue of a [[Hermitian matrix|0P17NGQQMQ]].\nA square-integrable Hermitian kernel gives a [[self-adjoint operator|0NEGN5CRSC]] on $L^2$, so its characteristic values are real and its eigenfunctions can be chosen orthonormal.\nThe Hilbert–Schmidt theory of integral equations is built on such kernels.\n\n## Explanations\n\n### L1 human\n\nSwapping the two variables of a Hermitian kernel gives its complex conjugate.\nFor a real kernel this simply means that the kernel is symmetric in its two variables.\nIt mirrors the rule $a_{ij} = \\overline{a_{ji}}$ for the entries of a Hermitian matrix.\nFor example, $K(x, s) = e^{i (x - s)}$ is Hermitian, while $K(x, s) = i$ is not.\n",
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