"코쉬 행렬과 행렬식"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) |
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| (사용자 2명의 중간 판 27개는 보이지 않습니다) | |||
| 1번째 줄: | 1번째 줄: | ||
| − | < | + | ==개요== |
| + | * 행렬 <math>A=({\frac{1}{x_i-y_j}})_{1\le i,j\le n}</math>를 크기 n인 코쉬 행렬이라 함 | ||
| + | * 행렬식 | ||
| + | :<math> | ||
| + | \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)} | ||
| + | </math> | ||
| + | :<math> | ||
| + | \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)} | ||
| + | </math> | ||
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| − | + | ==예== | |
| − | + | ===n=1인 경우=== | |
| + | * <math>\left( \begin{array}{c} \frac{1}{x_1-y_1} \end{array} \right)</math> | ||
| − | |||
| − | <math> | + | ===n=2인 경우=== |
| + | * 코쉬 행렬 | ||
| + | :<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> | ||
| + | * 행렬식 | ||
| + | :<math> | ||
| + | \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)} | ||
| + | </math> | ||
| − | + | ===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> | ||
| + | * 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> | ||
| − | + | ||
| − | <math>\left( \begin{array}{ | + | ===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> | ||
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| − | + | ==역사== | |
| + | * [[수학사 연표]] | ||
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| − | + | ||
| − | + | ==메모== | |
| + | * http://mathoverflow.net/questions/20609/what-role-does-cauchys-determinant-identity-play-in-combinatorics | ||
| − | + | ||
| − | * | + | ==관련된 항목들== |
| − | * | + | * [[반데몬드 행렬과 행렬식 (Vandermonde matrix)]] |
| + | * [[힐버트 행렬]] | ||
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| − | + | ==매스매티카 파일 및 계산 리소스== | |
| − | + | * https://docs.google.com/leaf?id=0B8XXo8Tve1cxM2E1ODYzMGUtYTJhMi00MmYxLWEzZDMtZDI2NmZmMWZmMDdm&sort=name&layout=list&num=50 | |
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| + | ==사전 형태의 자료== | ||
* http://en.wikipedia.org/wiki/Cauchy_matrix | * http://en.wikipedia.org/wiki/Cauchy_matrix | ||
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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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| − | + | [[분류:선형대수학]] | |
| + | [[분류:대칭다항식]] | ||
| + | [[분류:행렬식]] | ||
| − | + | ==메타데이터== | |
| − | + | ===위키데이터=== | |
| − | * | + | * ID : [https://www.wikidata.org/wiki/Q2915997 Q2915997] |
| − | + | ===Spacy 패턴 목록=== | |
| − | * [ | + | * [{'LOWER': 'cauchy'}, {'LEMMA': 'matrix'}] |
2021년 2월 17일 (수) 06:01 기준 최신판
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
메모
관련된 항목들
매스매티카 파일 및 계산 리소스
사전 형태의 자료
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.
메타데이터
위키데이터
- ID : Q2915997
Spacy 패턴 목록
- [{'LOWER': 'cauchy'}, {'LEMMA': 'matrix'}]