코쉬 행렬과 행렬식

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개요

  • 행렬 \(A=({\frac{1}{x_i-y_j}})_{1\le i,j\le n}\)를 크기 n인 코쉬 행렬이라 함
  • 행렬식

\[ \det \left(\frac{1}{x _i-y _j}\right) _{1\le i,j \le n}=(-1)^{\binom{n}{2}}\frac{\prod _{1\le i < j\le n} (x_j-x _i)(y _j-y _i)}{\prod _{i,j=1}^n (x _i-y _j)} \] \[ \det \left(\frac{1}{x _i+y _j}\right) _{1\le i,j \le n}=\frac{\prod _{1\le i < j\le n} (x_j-x _i)(y _j-y _i)}{\prod _{i,j=1}^n (x _i+y _j)} \]


n=1인 경우

  • \(\left( \begin{array}{c} \frac{1}{x_1-y_1} \end{array} \right)\)


n=2인 경우

  • 코쉬 행렬

\[\left( \begin{array}{cc} \frac{1}{x_1-y_1} & \frac{1}{x_1-y_2} \\ \frac{1}{x_2-y_1} & \frac{1}{x_2-y_2} \end{array} \right)\]

  • 행렬식

\[ \frac{\left(x_1-x_2\right) \left(y_1-y_2\right)}{\left(x_1-y_1\right) \left(y_1-x_2\right) \left(x_1-y_2\right) \left(x_2-y_2\right)} \]

n=3인 경우

  • 코쉬 행렬은

\[\left( \begin{array}{ccc} \frac{1}{x_1-y_1} & \frac{1}{x_1-y_2} & \frac{1}{x_1-y_3} \\ \frac{1}{x_2-y_1} & \frac{1}{x_2-y_2} & \frac{1}{x_2-y_3} \\ \frac{1}{x_3-y_1} & \frac{1}{x_3-y_2} & \frac{1}{x_3-y_3} \end{array} \right)\]

  • 행렬식은

\[-\frac{\left(-x_1+x_2\right) \left(-x_1+x_3\right) \left(-x_2+x_3\right) \left(y_1-y_2\right) \left(y_1-y_3\right) \left(y_2-y_3\right)}{\left(x_3-y_1\right) \left(-x_1+y_1\right) \left(-x_2+y_1\right) \left(x_2-y_2\right) \left(x_3-y_2\right) \left(-x_1+y_2\right) \left(x_1-y_3\right) \left(x_2-y_3\right) \left(x_3-y_3\right)}\]


n=4인 경우

  • 코쉬 행렬은

\[\left( \begin{array}{cccc} \frac{1}{x_1-y_1} & \frac{1}{x_1-y_2} & \frac{1}{x_1-y_3} & \frac{1}{x_1-y_4} \\ \frac{1}{x_2-y_1} & \frac{1}{x_2-y_2} & \frac{1}{x_2-y_3} & \frac{1}{x_2-y_4} \\ \frac{1}{x_3-y_1} & \frac{1}{x_3-y_2} & \frac{1}{x_3-y_3} & \frac{1}{x_3-y_4} \\ \frac{1}{x_4-y_1} & \frac{1}{x_4-y_2} & \frac{1}{x_4-y_3} & \frac{1}{x_4-y_4} \end{array} \right)\]



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관련논문

  • Ishikawa, Masao, Soichi Okada, Hiroyuki Tagawa, and Jiang Zeng. “Generalizations of Cauchy’s Determinant and Schur’s Pfaffian.” Advances in Applied Mathematics 36, no. 3 (2006): 251–87. doi:10.1016/j.aam.2005.07.001.
  • Chen, William Y. C., Christian Krattenthaler, and Arthur L. B. Yang. “The Flagged Cauchy Determinant.” Graphs and Combinatorics 21, no. 1 (2005): 51–62. doi:10.1007/s00373-004-0593-9.

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