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  "body": "\n## Summary\n\nA symmetric kernel is a real-valued [[kernel|GSMBXT7FPK]] with $K(x, s) = K(s, x)$ for all $x$ and $s$.\nIt is the continuous analogue of a [[symmetric matrix|0VR671PHE5]], and the real case of a [[Hermitian kernel|2ZB3NFZ94T]].\nA nonzero square-integrable symmetric kernel has real characteristic values and an orthonormal system of eigenfunctions, as the Hilbert–Schmidt theory shows.\nGreen's functions of self-adjoint boundary value problems are symmetric kernels.\n\n## Explanations\n\n### L1 human\n\nThe influence of the point $s$ on the point $x$ is the same as the influence of $x$ on $s$.\nIf one draws the kernel as a surface over the square, the surface is mirror symmetric across the diagonal $x = s$.\nFor example, $K(x, s) = \\min(x, s)$ and $K(x, s) = e^{-\\abs{x - s}}$ are symmetric.\nThe kernel $K(x, s) = x$ is not, because $K(1, 0) = 1$ but $K(0, 1) = 0$.\n",
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