# 希尔伯特矩阵

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

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

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

## 性质

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

${\displaystyle c_{n}=\prod _{i=1}^{n-1}i^{n-i}=\prod _{i=1}^{n-1}i!}$

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

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

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

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

## 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.