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  "body": "\n## Example\n\nFor $n \\ge 1$, the Hilbert matrix $H_n$ is the $n \\times n$ real matrix with entries\n$$(H_n)_{ij} = \\frac{1}{i + j - 1} \\qquad (1 \\le i, j \\le n).$$\nFor $n = 3$ it is\n$$H_3 = \\begin{pmatrix} 1 \u0026 \\tfrac12 \u0026 \\tfrac13 \\\\ \\tfrac12 \u0026 \\tfrac13 \u0026 \\tfrac14 \\\\ \\tfrac13 \u0026 \\tfrac14 \u0026 \\tfrac15 \\end{pmatrix}.$$\n\nIt is a [[Hankel matrix|PQ1D8FQB3E]]: the entry depends only on $i + j$, with $h_k = 1/(k-1)$.\n\nIt is a [[Cauchy matrix|95MTQMX59H]]: with $x_i = i$ and $y_j = 1 - j$, the entry is $1/(x_i - y_j)$, and all $x_i - y_j = i + j - 1$ are nonzero.\n\nIt is a [[positive definite matrix|0KX66CZMBB]].\nIndeed, $(H_n)_{ij} = \\int_0^1 t^{i-1} t^{j-1} \\, dt$, so $H_n$ is the [[Gram matrix|00SPJSDT3Z]] of the functions $1, t, \\dots, t^{n-1}$ for the inner product $\\ip{f}{g} = \\int_0^1 f(t) g(t) \\, dt$.\nFor $c \\in \\R^n$ and $p(t) = \\sum_i c_i t^{i-1}$, this gives $c^T H_n c = \\int_0^1 p(t)^2 \\, dt$, which is positive when $c \\ne 0$, because then $p$ is a nonzero polynomial.\n\n## Remarks\n\nThe inverse of $H_n$ has integer entries; for example $\\det H_2 = 1/12$ and $H_2^{-1} = \\begin{pmatrix} 4 \u0026 -6 \\\\ -6 \u0026 12 \\end{pmatrix}$.\nThe Hilbert matrix is a standard test case in numerical analysis, because it is badly conditioned: its condition number grows very fast with $n$.\n\n## Explanations\n\n### L1 human\n\nThe Hilbert matrix is a simple square table of fractions, $1$, $\\tfrac12$, $\\tfrac13$, and so on, arranged so that each anti-diagonal holds one fraction.\nIt comes from fitting a polynomial to a function by least squares on the interval from $0$ to $1$.\nIt looks harmless, but it is extremely close to a singular matrix, so computers lose many digits when they solve systems with it.\n",
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