코쉬 행렬과 행렬식

개요

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

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

예

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

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

역사

수학사 연표

메모

http://mathoverflow.net/questions/20609/what-role-does-cauchys-determinant-identity-play-in-combinatorics

매스매티카 파일 및 계산 리소스

https://docs.google.com/leaf?id=0B8XXo8Tve1cxM2E1ODYzMGUtYTJhMi00MmYxLWEzZDMtZDI2NmZmMWZmMDdm&sort=name&layout=list&num=50

사전 형태의 자료

http://en.wikipedia.org/wiki/Cauchy_matrix