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  "body": "\n## Statement\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]].\nLet $R_K$ be its [[Rayleigh quotient|HRCQ3JHYY0]] on $L^2[a, b]$.\nList the positive [[characteristic values|DC7HMHVREP]] of $K$ as $0 \u003c \\lambda_1 \\le \\lambda_2 \\le \\cdots$.\nEach $\\lambda$ is repeated as often as the dimension of the space of solutions of $\\varphi = \\lambda T_K \\varphi$.\nLet $k \\ge 1$ be such that there are at least $k$ of them.\nThen\n$$\\frac{1}{\\lambda_k} = \\max_{\\dim V = k} \\; \\min_{\\varphi \\in V \\setminus \\set{0}} R_K(\\varphi) = \\min_{\\psi_1, \\dots, \\psi_{k-1} \\in L^2[a, b]} \\; \\sup \\set{R_K(\\varphi) : \\varphi \\ne 0, \\ \\ip{\\varphi}{\\psi_j} = 0 \\text{ for } j \u003c k},$$\nwhere $V$ runs over the linear subspaces of $L^2[a, b]$ of dimension $k$.\n\n## Remarks\n\nLet $\\varphi_1, \\varphi_2, \\dots$ be orthonormal [[eigenfunctions|4NSACSWRJZ]] with $T_K \\varphi_j = \\varphi_j / \\lambda_j$, from the [[spectral theorem for compact self-adjoint operators|A1G25R99PY]].\nOn the span of $\\varphi_1, \\dots, \\varphi_k$, the quotient is at least $1/\\lambda_k$; so the max-min is at least $1/\\lambda_k$.\nAny $k$-dimensional $V$ contains a nonzero $\\varphi$ orthogonal to $\\varphi_1, \\dots, \\varphi_{k-1}$, and on that orthogonal complement $R_K \\le 1/\\lambda_k$; so the max-min is exactly $1/\\lambda_k$.\nThe min-max form is proved in the same way, with the choice $\\psi_j = \\varphi_j$ attaining the minimum.\n\nExample: for $K(x, y) = \\min(x, y)$ on $[0, 1]$, $\\lambda_k = (k - \\tfrac12)^2 \\pi^2$.\nFor $k = 1$ the theorem says $\\int_0^1 \\int_0^1 \\min(x, y) \\varphi(x) \\varphi(y) \\, dx \\, dy \\le \\frac{4}{\\pi^2} \\int_0^1 \\varphi^2$ for real $\\varphi$, with equality for $\\varphi(x) = \\sin(\\pi x / 2)$.\n\nThe symmetry of $K$ is needed: without it the quotient is not real, and the characteristic values need not be real or exist at all.\nThe negative characteristic values satisfy the same formulas applied to the kernel $-K$.\n\n## Explanations\n\n### L1 human\n\nThe characteristic values of a symmetric kernel can be found by optimization.\nThe smallest positive one corresponds to the largest stretching of a single function by the integral operator.\nThe next ones are the best stretching that one can guarantee when some directions are forbidden.\n\n### L2 human\n\nThis is the integral form of the [[Courant–Fischer min-max theorem|CJK2HS2508]] for Hermitian matrices.\nIt allows one to compare characteristic values of two kernels without computing them: if $R_{K_1} \\le R_{K_2}$ everywhere, then the $k$-th positive characteristic value of $K_1$ is at least that of $K_2$.\nIt is the basis of the Rayleigh–Ritz method, which gives bounds for characteristic values by restricting the quotient to finite-dimensional trial spaces.\n",
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