힐베르트 행렬

개요

• 코쉬 행렬의 특별한 경우
• 항켈 행렬의 예
• 크기 $n$인 힐베르트 행렬 $H=(H_{ij})_{1\leq i,j\leq n}$의 성분은 \(H_{ij} = \frac{1}{i+j-1}\)로 주어진다

예

$$ \left( \begin{array}{c} 1 \\ \end{array} \right) $$

$$ \left( \begin{array}{cc} 1 & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{3} \\ \end{array} \right) $$

$$ \left( \begin{array}{ccc} 1 & \frac{1}{2} & \frac{1}{3} \\ \frac{1}{2} & \frac{1}{3} & \frac{1}{4} \\ \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\ \end{array} \right) $$

$$ \left( \begin{array}{cccc} 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} \\ \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\ \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} \\ \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} \\ \end{array} \right) $$

$$ \left( \begin{array}{ccccc} 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\ \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} \\ \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} \\ \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} \\ \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} & \frac{1}{9} \\ \end{array} \right) $$

행렬식

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

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

관련된 항목들

• Glaisher–Kinkelin 상수

매스매티카 파일 및 계산 리소스

• https://docs.google.com/leaf?id=0B8XXo8Tve1cxZWU3NWQ5ZGYtNWUzZC00NzEyLTgwZGUtNmMwZjEzMmVmOGIx&sort=name&layout=list&num=50

사전 형태의 자료

• http://en.wikipedia.org/wiki/Hilbert_matrix
• http://mathworld.wolfram.com/HilbertMatrix.html

관련논문

• Choi, M.-D. "Tricks or Treats with the Hilbert Matrix." Amer. Math. Monthly 90, 301-312, 1983.