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  "body": "\n## Summary\n\nA characteristic value of a kernel $K$ is a number $\\lambda$ for which the homogeneous integral equation $\\varphi(x) = \\lambda \\int K(x, y) \\varphi(y) \\, dy$ has a nonzero solution.\nIt is the reciprocal of a nonzero [[eigenvalue|0BQAA2BK3Q]] of the integral operator with kernel $K$.\nCharacteristic values are the exceptional parameters of the [[Fredholm equation of the second kind|DEYYNVCV9J]].\nAway from them, the equation has exactly one solution for each right-hand side.\n\n## Explanations\n\n### L1 human\n\nAn integral equation of the second kind contains a parameter $\\lambda$ in front of the integral.\nFor most values of $\\lambda$ the equation is well behaved, but for some special values the homogeneous equation has a nonzero solution, and uniqueness fails.\nThese special values are the characteristic values.\nFor $K(x, y) = xy$ on $[0, 1]$, the only one is $\\lambda = 3$, with solution $\\varphi(x) = x$, because $\\int_0^1 x y \\cdot y \\, dy = x/3$.\n",
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