# 希尔伯特矩阵

$H_{ij} = \frac{1}{i+j-1}.$

$H_5 = \begin{bmatrix} 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\[4pt] \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} \\[4pt] \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} \\[4pt] \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} \\[4pt] \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} & \frac{1}{9} \end{bmatrix}.$

$H_{ij} = \int_{0}^{1} x^{i+j-2} \, dx,$

## 性质

$\det(H)={{c_n^{\;4}}\over {c_{2n}}}$

$c_n = \prod_{i=1}^{n-1} i^{n-i}=\prod_{i=1}^{n-1} i!$

${1 \over \det (H)}={{c_{2n}}\over {c_n^{\;4}}}=n!\cdot \prod_{i=1}^{2n-1} {i \choose [i/2]}$

$\det(H)=a_n\, n^{-1/4}(2\pi)^n \,4^{-n^2}$

$(H^{-1})_{ij}=(-1)^{i+j}(i+j-1){n+i-1 \choose n-j}{n+j-1 \choose n-i}{i+j-2 \choose i-1}^2$

$n\rightarrow\infty$ 的时候，$n \times n$的希尔伯特矩阵的条件数近似为$O((1+\sqrt{2})^{4n}/\sqrt{n})$

## 参考来源

• David Hilbert, Collected papers, vol. II, article 21.
• Beckermann, Bernhard. "The condition number of real Vandermonde, Krylov and positive definite Hankel matrices" in Numerische Mathematik. 85(4), 553--577, 2000.
• Choi, M.-D. "Tricks or Treats with the Hilbert Matrix" in American Mathematical Monthly. 90, 301–312, 1983.
• Todd, John. "The Condition Number of the Finite Segment of the Hilbert Matrix" in National Bureau of Standards, Applied Mathematics Series. 39, 109–116, 1954.
• Wilf, H.S. Finite Sections of Some Classical Inequalities. Heidelberg: Springer, 1970.