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  "body": "\n## Definition\n\nLet $a \u003c b$ and let $K \\in L^2([a, b] \\times [a, b])$ be a [[square-integrable kernel|WWMWX7AD4V]] that is [[Hermitian|2ZB3NFZ94T]]: $K(y, x) = \\overline{K(x, y)}$ for almost all $x, y$.\nLet $T_K$ be its [[Hilbert–Schmidt integral operator|CB5G212Q03]] on $L^2[a, b]$, and let $\\ip{u}{v} = \\int_a^b u(x) \\overline{v(x)} \\, dx$.\nThe *Rayleigh quotient* of $K$ at a function $\\varphi \\in L^2[a, b]$ that is not $0$ almost everywhere is\n$$R_K(\\varphi) = \\frac{\\ip{T_K \\varphi}{\\varphi}}{\\ip{\\varphi}{\\varphi}} = \\frac{\\int_a^b \\int_a^b K(x, y) \\, \\varphi(y) \\, \\overline{\\varphi(x)} \\, dx \\, dy}{\\int_a^b \\abs{\\varphi(x)}^2 \\, dx}.$$\n\n## Remarks\n\nSince $T_K$ is self-adjoint, $R_K(\\varphi)$ is a real number, and $R_K(c \\varphi) = R_K(\\varphi)$ for every nonzero $c \\in \\C$.\nIf $\\varphi$ is an eigenfunction with eigenvalue $\\mu$, then $R_K(\\varphi) = \\mu$.\n\nExample: for $K(x, y) = xy$ on $[0, 1]$ and $\\varphi = 1$, the numerator is $\\int_0^1 \\int_0^1 xy \\, dx \\, dy = 1/4$ and the denominator is $1$, so $R_K(1) = 1/4$.\nFor $\\varphi(x) = x$, the numerator is $\\left(\\int_0^1 x^2 \\, dx\\right)^2 = 1/9$ and the denominator is $1/3$, so $R_K(\\varphi) = 1/3$, the eigenvalue of $x$.\n\nNon-example: the constant kernel $K(x, y) = i$ is not Hermitian, since $\\overline{K(x, y)} = -i \\ne K(y, x)$.\nThe same formula at $\\varphi = 1$ on $[0, 1]$ gives $i$, which is not real, so it is not a Rayleigh quotient in the sense above.\n\nFor a Hermitian matrix $A$, the same formula with sums instead of integrals is the [[Rayleigh quotient|06WHP8SE98]] $x^* A x / x^* x$.\n",
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