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  "body": "\n## Summary\n\nLet $K$ be a Hermitian kernel with integral operator $T_K$.\nThe Rayleigh quotient of $K$ at a nonzero function $\\varphi$ is the real number $\\ip{T_K \\varphi}{\\varphi} / \\ip{\\varphi}{\\varphi}$.\nIt is the integral analogue of the [[Rayleigh quotient|06WHP8SE98]] of a Hermitian matrix.\nIts extreme values and critical points give the eigenvalues and eigenfunctions of $T_K$, which turns eigenvalue problems for integral equations into problems of optimization.\n\n## Explanations\n\n### L1 human\n\nThe Rayleigh quotient measures how much the integral operator stretches a function in its own direction.\nThe function that is stretched the most is an eigenfunction, and the amount of stretching is the largest eigenvalue.\nFor $K(x, y) = xy$ on $[0, 1]$ and $\\varphi = 1$, the quotient is $1/4$, while for the eigenfunction $\\varphi(x) = x$ it reaches the maximum $1/3$.\n",
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