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  "body": "\n## Definition\n\nLet $H$ be a real or complex [[Hilbert space|HM2HHE3J51]] with inner product $\\ip{\\cdot}{\\cdot}$.\nLet $A$ be a [[self-adjoint operator|0NEGN5CRSC]] in $H$ that is bounded below.\nLet $Q$ be its closed quadratic form with form domain $D(Q)$, so that $Q(u, v) = \\ip{Au}{v}$ for $u$ in the domain of $A$.\nLet $m \\ge 1$ and let $\\varphi_1, \\dots, \\varphi_m \\in D(Q)$ be linearly independent trial vectors.\nLet $K$ and $M$ be the $m \\times m$ matrices with entries $K_{ij} = Q(\\varphi_j, \\varphi_i)$ and $M_{ij} = \\ip{\\varphi_j}{\\varphi_i}$.\nThe *Rayleigh–Ritz method* computes the solutions of the generalised eigenvalue problem\n$$K c = \\mu M c, \\qquad c \\in \\C^m \\setminus \\set{0}.$$\nIts eigenvalues $\\mu_1 \\le \\cdots \\le \\mu_m$ are the *Ritz values*, and the vectors $\\sum_j c_j \\varphi_j$ for the eigenvectors $c$ are the *Ritz vectors*.\n\n## Remarks\n\nThe Ritz values are the stationary values of the [[Rayleigh quotient|06WHP8SE98]] $Q(u, u) / \\ip{u}{u}$ on the span $V$ of the trial vectors.\nIf the spectrum of $A$ below the essential spectrum consists of [[eigenvalues|0BQAA2BK3Q]] $\\lambda_1 \\le \\lambda_2 \\le \\cdots$, the [[min-max theorem|G11NE3AT6E]] gives $\\lambda_k \\le \\mu_k$ for $k = 1, \\dots, m$.\nFor a Hermitian $n \\times n$ matrix $A$ and a matrix $W$ with $m$ orthonormal columns, the Ritz values are the eigenvalues of $W^* A W$.\n\nExample: $H = L^2(0, 1)$ and $A = -\\frac{d^2}{dx^2}$ with Dirichlet conditions, so $D(Q) = H^1_0(0, 1)$, $Q(u, u) = \\int_0^1 \\abs{u'}^2 \\, dx$ and $\\lambda_1 = \\pi^2$.\nWith $m = 1$ and $\\varphi_1(x) = x (1 - x)$, which vanishes at $0$ and $1$, one gets $K = \\int_0^1 (1 - 2x)^2 \\, dx = \\frac{1}{3}$ and $M = \\int_0^1 x^2 (1 - x)^2 \\, dx = \\frac{1}{30}$.\nSo $\\mu_1 = 10$, an upper bound for $\\pi^2 \\approx 9.8696$.\n\nNon-example: the trial function $\\varphi_1 = 1$ is not in $H^1_0(0, 1)$, because it does not vanish at the ends.\nIts Rayleigh quotient is $0 \u003c \\pi^2$, so the upper bound fails when the trial vectors leave the form domain.\n\n## Explanations\n\n### L1 human\n\nThe method restricts the eigenvalue problem to a few trial vectors and solves the resulting small matrix problem, which gives upper bounds for the lowest eigenvalues.\n",
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