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  "body": "\n## Summary\n\nAn eigenfunction of an integral operator $T_K$ is a nonzero function $\\varphi$ with $T_K \\varphi = \\mu \\varphi$ for some number $\\mu$.\nIt is an [[eigenvector|0N6960W0BV]] of $T_K$ in a space of functions.\nFor symmetric kernels the eigenfunctions can be chosen to form an orthonormal system, and expanding in them diagonalizes the operator, as in the [[Hilbert–Schmidt theorem|A1G25R99PY]].\n\n## Explanations\n\n### L1 human\n\nApplying the integral operator to an eigenfunction does not change its shape; it only multiplies it by a number.\nSo eigenfunctions are the natural modes of the operator, like the pure tones of a vibrating string.\nFor the kernel $K(x, y) = \\min(x, y)$ on $[0, 1]$, the function $\\sin(\\pi x / 2)$ is an eigenfunction with eigenvalue $4 / \\pi^2$.\n",
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