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  "body": "\n## Definition\n\nLet $a \u003c b$, let $K \\in L^2([a, b] \\times [a, b])$, and let $T_K$ be the [[Hilbert–Schmidt integral operator|CB5G212Q03]] $(T_K \\varphi)(x) = \\int_a^b K(x, y) \\varphi(y) \\, dy$ on $L^2[a, b]$.\nAn *eigenfunction* of $T_K$ is a function $\\varphi \\in L^2[a, b]$, not $0$ almost everywhere, such that\n$$\\int_a^b K(x, y) \\, \\varphi(y) \\, dy = \\mu \\, \\varphi(x) \\quad \\text{for almost every } x \\in [a, b]$$\nfor some $\\mu \\in \\C$.\nThe number $\\mu$ is the [[eigenvalue|0BQAA2BK3Q]] of $\\varphi$.\n\n## Remarks\n\nSo an eigenfunction is an [[eigenvector|0N6960W0BV]] of $T_K$ in the vector space $L^2[a, b]$.\nWhen $\\mu \\ne 0$, the eigenfunctions for $\\mu$ are the nonzero solutions of the homogeneous equation $\\varphi = \\lambda T_K \\varphi$ with $\\lambda = 1/\\mu$.\n\nExample: for $K(x, y) = \\min(x, y)$ on $[0, 1]$ and $\\varphi(x) = \\sin(\\pi x / 2)$, one has $\\int_0^1 \\min(x, y) \\varphi(y) \\, dy = \\frac{4}{\\pi^2} \\varphi(x)$.\nTo check, call the left side $g(x)$; then $g(0) = 0$, $g'(1) = 0$ and $g'' = -\\varphi$, and $\\frac{4}{\\pi^2} \\varphi$ satisfies the same three conditions, which determine $g$.\n\nNon-example: for the same kernel, $\\varphi = 1$ is not an eigenfunction, since $\\int_0^1 \\min(x, y) \\, dy = x - x^2/2$ is not a constant multiple of $1$.\n\nThe same definition is used with $L^2[a, b]$ replaced by $C[a, b]$ for continuous kernels, or by the $L^2$ space of a measure space.\nFor continuous kernels and $\\mu \\ne 0$, every eigenfunction in $L^2$ agrees almost everywhere with a continuous one.\n",
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