"코쉬 행렬과 행렬식"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) |
Pythagoras0 (토론 | 기여) |
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| 1번째 줄: | 1번째 줄: | ||
| − | == | + | ==개요== |
| − | + | * 행렬 $A=(a_{i,j})_{1\ge i,j\le n}를 크기 n인 코쉬 행렬이라 함. 여기서 | |
| − | * | + | :<math>a_{ij}={\frac{1}{x_i-y_j}}</math> |
| − | + | * 행렬식의 계산 | |
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| − | <math> | + | ==n=1인 경우== |
| + | * <math>\left( \begin{array}{c} \frac{1}{x_1-y_1} \end{array} \right)</math> | ||
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| − | <math>\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)</math> | + | ==n=2인 경우== |
| + | * <math>\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)</math> | ||
==n=3인 경우== | ==n=3인 경우== | ||
| − | + | * 코쉬 행렬은 | |
| − | <math>\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)</math> | + | :<math>\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)</math> |
| − | + | * 행렬식은 | |
| − | 행렬식은 | + | :<math>-\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)}</math> |
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| − | <math>-\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)}</math> | ||
| − | n=4인 경우 | + | ==n=4인 경우== |
| − | + | * 코쉬 행렬은 | |
| − | <math>\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)</math> | + | :<math>\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)</math> |
| 36번째 줄: | 32번째 줄: | ||
==역사== | ==역사== | ||
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* [[수학사 연표]] | * [[수학사 연표]] | ||
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| 55번째 줄: | 45번째 줄: | ||
==관련된 항목들== | ==관련된 항목들== | ||
| − | * [[힐버트 행렬]] | + | * [[힐버트 행렬]] |
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* https://docs.google.com/leaf?id=0B8XXo8Tve1cxM2E1ODYzMGUtYTJhMi00MmYxLWEzZDMtZDI2NmZmMWZmMDdm&sort=name&layout=list&num=50 | * https://docs.google.com/leaf?id=0B8XXo8Tve1cxM2E1ODYzMGUtYTJhMi00MmYxLWEzZDMtZDI2NmZmMWZmMDdm&sort=name&layout=list&num=50 | ||
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* [[매스매티카 파일 목록]] | * [[매스매티카 파일 목록]] | ||
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==사전 형태의 자료== | ==사전 형태의 자료== | ||
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* http://en.wikipedia.org/wiki/Cauchy_matrix | * http://en.wikipedia.org/wiki/Cauchy_matrix | ||
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[[분류:선형대수학]] | [[분류:선형대수학]] | ||
2013년 2월 14일 (목) 15:15 판
개요
- 행렬 $A=(a_{i,j})_{1\ge i,j\le n}를 크기 n인 코쉬 행렬이라 함. 여기서
\[a_{ij}={\frac{1}{x_i-y_j}}\]
- 행렬식의 계산
n=1인 경우
- \(\left( \begin{array}{c} \frac{1}{x_1-y_1} \end{array} \right)\)
n=2인 경우
- \(\left( \begin{array}{cc} \frac{1}{x_1-y_1} & \frac{1}{x_1-y_2} \\ \frac{1}{x_2-y_1} & \frac{1}{x_2-y_2} \end{array} \right)\)
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
메모
관련된 항목들
매스매티카 파일 및 계산 리소스
- https://docs.google.com/leaf?id=0B8XXo8Tve1cxM2E1ODYzMGUtYTJhMi00MmYxLWEzZDMtZDI2NmZmMWZmMDdm&sort=name&layout=list&num=50
- 매스매티카 파일 목록