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    Hilbert Matrix

    Entries

    H_n_(i j) = 1/(i + j - 1)

    Low-dimensional Hilbert matrices

    rank | Hilbert matrix 1 | (1) 2 | (1 | 1/2 1/2 | 1/3) 3 | (1 | 1/2 | 1/3 1/2 | 1/3 | 1/4 1/3 | 1/4 | 1/5) 4 | (1 | 1/2 | 1/3 | 1/4 1/2 | 1/3 | 1/4 | 1/5 1/3 | 1/4 | 1/5 | 1/6 1/4 | 1/5 | 1/6 | 1/7)

    Example matrix plot

    Example matrix plot

    Example eigenvalue plot

    Example eigenvalue plot

    Example eigenvector plot

    Example eigenvector plot

    Determinant

    left bracketing bar H_n right bracketing bar = 1/(n! product_(i=1)^(2 n - 1) binomial(i, floor(i/2)))

    left bracketing bar H_n right bracketing bar = (2^(n (2 n - 1) - 1/12) A^3 G(n + 1/2) G(n + 3/2))/((e^(1/4) π^n) G(n + 1)^2)

    lim_(n->∞) (4^(n^2) n^(1/4) left bracketing bar H_n right bracketing bar )/(2 π)^n = (e^(1/4) 2^(1/12))/A^3

    Inverse

    H_n^(-1)_(i j) = (-1)^(i + j) (i + j - 1) binomial(n + i - 1, n - j) binomial(n + j - 1, n - i) binomial(i + j - 2, i - 1)^2

    Properties

    symmetric | positive definite | totally positive | Hankel | compact

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