"파피안(Pfaffian)"의 두 판 사이의 차이
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==교대행렬과 행렬식== | ==교대행렬과 행렬식== | ||
| − | * 2×2 교대행렬:<math>\left( \begin{array}{cc} 0 & t_{1,2} \\ -t_{1,2} & 0 \end{array} \right)</math> 의 행렬식 <math>t_{1,2}^2</math | + | * 2×2 교대행렬:<math>\left( \begin{array}{cc} 0 & t_{1,2} \\ -t_{1,2} & 0 \end{array} \right)</math> 의 행렬식 <math>t_{1,2}^2</math> |
| − | * 4×4 교대행렬:<math>\left( \begin{array}{cccc} 0 & t_{1,2} & t_{1,3} & t_{1,4} \\ -t_{1,2} & 0 & t_{2,3} & t_{2,4} \\ -t_{1,3} & -t_{2,3} & 0 & t_{3,4} \\ -t_{1,4} & -t_{2,4} & -t_{3,4} & 0 \end{array} \right)</math>, 행렬식 <math>\left(t_{1,4} t_{2,3}-t_{1,3} t_{2,4}+t_{1,2} t_{3,4}\right){}^2</math | + | * 4×4 교대행렬:<math>\left( \begin{array}{cccc} 0 & t_{1,2} & t_{1,3} & t_{1,4} \\ -t_{1,2} & 0 & t_{2,3} & t_{2,4} \\ -t_{1,3} & -t_{2,3} & 0 & t_{3,4} \\ -t_{1,4} & -t_{2,4} & -t_{3,4} & 0 \end{array} \right)</math>, 행렬식 <math>\left(t_{1,4} t_{2,3}-t_{1,3} t_{2,4}+t_{1,2} t_{3,4}\right){}^2</math> |
| − | * 6×6 교대행렬:<math>\left( \begin{array}{cccccc} 0 & t_{1,2} & t_{1,3} & t_{1,4} & t_{1,5} & t_{1,6} \\ -t_{1,2} & 0 & t_{2,3} & t_{2,4} & t_{2,5} & t_{2,6} \\ -t_{1,3} & -t_{2,3} & 0 & t_{3,4} & t_{3,5} & t_{3,6} \\ -t_{1,4} & -t_{2,4} & -t_{3,4} & 0 & t_{4,5} & t_{4,6} \\ -t_{1,5} & -t_{2,5} & -t_{3,5} & -t_{4,5} & 0 & t_{5,6} \\ -t_{1,6} & -t_{2,6} & -t_{3,6} & -t_{4,6} & -t_{5,6} & 0 \end{array} \right)</math>, | + | * 6×6 교대행렬:<math>\left( \begin{array}{cccccc} 0 & t_{1,2} & t_{1,3} & t_{1,4} & t_{1,5} & t_{1,6} \\ -t_{1,2} & 0 & t_{2,3} & t_{2,4} & t_{2,5} & t_{2,6} \\ -t_{1,3} & -t_{2,3} & 0 & t_{3,4} & t_{3,5} & t_{3,6} \\ -t_{1,4} & -t_{2,4} & -t_{3,4} & 0 & t_{4,5} & t_{4,6} \\ -t_{1,5} & -t_{2,5} & -t_{3,5} & -t_{4,5} & 0 & t_{5,6} \\ -t_{1,6} & -t_{2,6} & -t_{3,6} & -t_{4,6} & -t_{5,6} & 0 \end{array} \right)</math>, 행렬식 <math>\left(t_{1,6} t_{2,5} t_{3,4}-t_{1,5} t_{2,6} t_{3,4}-t_{1,6} t_{2,4} t_{3,5}+t_{1,4} t_{2,6} t_{3,5}+t_{1,5} t_{2,4} t_{3,6}-t_{1,4} t_{2,5} t_{3,6}+t_{1,6} t_{2,3} t_{4,5}-t_{1,3} t_{2,6} t_{4,5}+t_{1,2} t_{3,6} t_{4,5}-t_{1,5} t_{2,3} t_{4,6}+t_{1,3} t_{2,5} t_{4,6}-t_{1,2} t_{3,5} t_{4,6}+t_{1,4} t_{2,3} t_{5,6}-t_{1,3} t_{2,4} t_{5,6}+t_{1,2} t_{3,4} t_{5,6}\right){}^2</math> |
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==파피안== | ==파피안== | ||
| − | * <math>A=(t_{i,j})</math> 로 주어진 교대행렬에 대하여 파피안을 다음과 같이 정의함:<math>\operatorname{pf}(A) = \frac{1}{2^n n!}\sum_{\sigma\in S_{2n}}\operatorname{sgn}(\sigma)\prod_{i=1}^{n}t_{\sigma(2i-1),\sigma(2i)}</math | + | * <math>A=(t_{i,j})</math> 로 주어진 교대행렬에 대하여 파피안을 다음과 같이 정의함:<math>\operatorname{pf}(A) = \frac{1}{2^n n!}\sum_{\sigma\in S_{2n}}\operatorname{sgn}(\sigma)\prod_{i=1}^{n}t_{\sigma(2i-1),\sigma(2i)}</math> |
| − | * n=1인 경우:<math>t_{1,2}</math | + | * n=1인 경우:<math>t_{1,2}</math> |
| − | * n=2인 경우:<math>t_{1,4} t_{2,3}-t_{1,3} t_{2,4}+t_{1,2} t_{3,4}</math | + | * n=2인 경우:<math>t_{1,4} t_{2,3}-t_{1,3} t_{2,4}+t_{1,2} t_{3,4}</math> |
| − | * n=3 인 경우:<math>t_{1,6} t_{2,5} t_{3,4}-t_{1,5} t_{2,6} t_{3,4}-t_{1,6} t_{2,4} t_{3,5}+t_{1,4} t_{2,6} t_{3,5}+t_{1,5} t_{2,4} t_{3,6}-t_{1,4} t_{2,5} t_{3,6}+t_{1,6} t_{2,3} t_{4,5}-t_{1,3} t_{2,6} t_{4,5}+t_{1,2} t_{3,6} t_{4,5}-t_{1,5} t_{2,3} t_{4,6}+t_{1,3} t_{2,5} t_{4,6}-t_{1,2} t_{3,5} t_{4,6}+t_{1,4} t_{2,3} t_{5,6}-t_{1,3} t_{2,4} t_{5,6}+t_{1,2} t_{3,4} t_{5,6}</math | + | * n=3 인 경우:<math>t_{1,6} t_{2,5} t_{3,4}-t_{1,5} t_{2,6} t_{3,4}-t_{1,6} t_{2,4} t_{3,5}+t_{1,4} t_{2,6} t_{3,5}+t_{1,5} t_{2,4} t_{3,6}-t_{1,4} t_{2,5} t_{3,6}+t_{1,6} t_{2,3} t_{4,5}-t_{1,3} t_{2,6} t_{4,5}+t_{1,2} t_{3,6} t_{4,5}-t_{1,5} t_{2,3} t_{4,6}+t_{1,3} t_{2,5} t_{4,6}-t_{1,2} t_{3,5} t_{4,6}+t_{1,4} t_{2,3} t_{5,6}-t_{1,3} t_{2,4} t_{5,6}+t_{1,2} t_{3,4} t_{5,6}</math> |
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==메모== | ==메모== | ||
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* http://www.science.uva.nl/onderwijs/thesis/centraal/files/f887198315.pdf | * http://www.science.uva.nl/onderwijs/thesis/centraal/files/f887198315.pdf | ||
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==매스매티카 파일 및 계산 리소스== | ==매스매티카 파일 및 계산 리소스== | ||
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* http://en.wikipedia.org/wiki/Talk%3APfaffian#Mathematica_code | * http://en.wikipedia.org/wiki/Talk%3APfaffian#Mathematica_code | ||
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==관련된 항목들== | ==관련된 항목들== | ||
| 66번째 줄: | 55번째 줄: | ||
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| − | ==사전 | + | ==사전 형태의 자료== |
* http://en.wikipedia.org/wiki/Pfaffian | * http://en.wikipedia.org/wiki/Pfaffian | ||
* http://en.wikipedia.org/wiki/FKT_algorithm | * http://en.wikipedia.org/wiki/FKT_algorithm | ||
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==리뷰, 에세이, 강의노트== | ==리뷰, 에세이, 강의노트== | ||
* Hirota, Ryogo. "Determinants and Pfaffians." 数理解析研究所講究録 1302 (2003): 220-242. http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1302-14.pdf | * Hirota, Ryogo. "Determinants and Pfaffians." 数理解析研究所講究録 1302 (2003): 220-242. http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1302-14.pdf | ||
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==관련논문== | ==관련논문== | ||
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* Wu, F. Y. 2006. “Pfaffian solution of a dimer-monomer problem: Single monomer on the boundary.” <em>Physical Review E</em> 74 (2): 020104. doi:10.1103/PhysRevE.74.020104. | * Wu, F. Y. 2006. “Pfaffian solution of a dimer-monomer problem: Single monomer on the boundary.” <em>Physical Review E</em> 74 (2): 020104. doi:10.1103/PhysRevE.74.020104. | ||
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==관련도서== | ==관련도서== | ||
| − | + | * Hirota, Ryogo, Atsushi Nagai, Jon Nimmo, and Claire Gilson. 2004. “Determinants and Pfaffians.” In The Direct Method in Soliton Theory. Cambridge Tracts in Mathematics. http://dx.doi.org/10.1017/CBO9780511543043.004. | |
* Barry M McCoy, Advanced Statistical Mechanics | * Barry M McCoy, Advanced Statistical Mechanics | ||
** The Pfaffian solution of the Ising model DOI:10.1093/acprof:oso/9780199556632.003.0011 | ** The Pfaffian solution of the Ising model DOI:10.1093/acprof:oso/9780199556632.003.0011 | ||
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[[분류:선형대수학]] | [[분류:선형대수학]] | ||
| + | [[분류:행렬식]] | ||
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| + | ==메타데이터== | ||
| + | ===위키데이터=== | ||
| + | * ID : [https://www.wikidata.org/wiki/Q1189744 Q1189744] | ||
| + | ===Spacy 패턴 목록=== | ||
| + | * [{'LEMMA': 'pfaffian'}] | ||
2021년 2월 17일 (수) 05:05 기준 최신판
개요
- 교대행렬(alternating matrix, 또는 skew-symmetric matrix)의 행렬식은 어떤 다항식의 제곱이 되는 성질을 가진다
- 교대행렬에 대해, 이 행렬식의 제곱근의 하나를 파피안으로 정의한다.
- \( \operatorname{pf(A)}^2=\operatorname{det(A)}\)
- \(\operatorname{pf}(BAB^T)= \det(B)\operatorname{pf}(A)\)
교대행렬과 행렬식
- 2×2 교대행렬\[\left( \begin{array}{cc} 0 & t_{1,2} \\ -t_{1,2} & 0 \end{array} \right)\] 의 행렬식 \(t_{1,2}^2\)
- 4×4 교대행렬\[\left( \begin{array}{cccc} 0 & t_{1,2} & t_{1,3} & t_{1,4} \\ -t_{1,2} & 0 & t_{2,3} & t_{2,4} \\ -t_{1,3} & -t_{2,3} & 0 & t_{3,4} \\ -t_{1,4} & -t_{2,4} & -t_{3,4} & 0 \end{array} \right)\], 행렬식 \(\left(t_{1,4} t_{2,3}-t_{1,3} t_{2,4}+t_{1,2} t_{3,4}\right){}^2\)
- 6×6 교대행렬\[\left( \begin{array}{cccccc} 0 & t_{1,2} & t_{1,3} & t_{1,4} & t_{1,5} & t_{1,6} \\ -t_{1,2} & 0 & t_{2,3} & t_{2,4} & t_{2,5} & t_{2,6} \\ -t_{1,3} & -t_{2,3} & 0 & t_{3,4} & t_{3,5} & t_{3,6} \\ -t_{1,4} & -t_{2,4} & -t_{3,4} & 0 & t_{4,5} & t_{4,6} \\ -t_{1,5} & -t_{2,5} & -t_{3,5} & -t_{4,5} & 0 & t_{5,6} \\ -t_{1,6} & -t_{2,6} & -t_{3,6} & -t_{4,6} & -t_{5,6} & 0 \end{array} \right)\], 행렬식 \(\left(t_{1,6} t_{2,5} t_{3,4}-t_{1,5} t_{2,6} t_{3,4}-t_{1,6} t_{2,4} t_{3,5}+t_{1,4} t_{2,6} t_{3,5}+t_{1,5} t_{2,4} t_{3,6}-t_{1,4} t_{2,5} t_{3,6}+t_{1,6} t_{2,3} t_{4,5}-t_{1,3} t_{2,6} t_{4,5}+t_{1,2} t_{3,6} t_{4,5}-t_{1,5} t_{2,3} t_{4,6}+t_{1,3} t_{2,5} t_{4,6}-t_{1,2} t_{3,5} t_{4,6}+t_{1,4} t_{2,3} t_{5,6}-t_{1,3} t_{2,4} t_{5,6}+t_{1,2} t_{3,4} t_{5,6}\right){}^2\)
파피안
- \(A=(t_{i,j})\) 로 주어진 교대행렬에 대하여 파피안을 다음과 같이 정의함\[\operatorname{pf}(A) = \frac{1}{2^n n!}\sum_{\sigma\in S_{2n}}\operatorname{sgn}(\sigma)\prod_{i=1}^{n}t_{\sigma(2i-1),\sigma(2i)}\]
- n=1인 경우\[t_{1,2}\]
- n=2인 경우\[t_{1,4} t_{2,3}-t_{1,3} t_{2,4}+t_{1,2} t_{3,4}\]
- n=3 인 경우\[t_{1,6} t_{2,5} t_{3,4}-t_{1,5} t_{2,6} t_{3,4}-t_{1,6} t_{2,4} t_{3,5}+t_{1,4} t_{2,6} t_{3,5}+t_{1,5} t_{2,4} t_{3,6}-t_{1,4} t_{2,5} t_{3,6}+t_{1,6} t_{2,3} t_{4,5}-t_{1,3} t_{2,6} t_{4,5}+t_{1,2} t_{3,6} t_{4,5}-t_{1,5} t_{2,3} t_{4,6}+t_{1,3} t_{2,5} t_{4,6}-t_{1,2} t_{3,5} t_{4,6}+t_{1,4} t_{2,3} t_{5,6}-t_{1,3} t_{2,4} t_{5,6}+t_{1,2} t_{3,4} t_{5,6}\]
메모
- number of perfect matchings on a planar rectangular lattice
- every non-zero term in the Pfaffian of the adjacency matrix of a graph G corresponds to a perfect matching.
- 통계물리에서 중요한 역할
- 도미노 타일링
- 다이머 모델
- http://www.science.uva.nl/onderwijs/thesis/centraal/files/f887198315.pdf
매스매티카 파일 및 계산 리소스
- https://docs.google.com/leaf?id=0B8XXo8Tve1cxN2RmNDAyYzItYjlmYy00MzM5LWJkZmQtYjdjOWZhNjM3MTI0&sort=name&layout=list&num=50
- http://en.wikipedia.org/wiki/Talk%3APfaffian#Mathematica_code
관련된 항목들
사전 형태의 자료
리뷰, 에세이, 강의노트
- Hirota, Ryogo. "Determinants and Pfaffians." 数理解析研究所講究録 1302 (2003): 220-242. http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1302-14.pdf
관련논문
- Wu, F. Y. 2006. “Pfaffian solution of a dimer-monomer problem: Single monomer on the boundary.” Physical Review E 74 (2): 020104. doi:10.1103/PhysRevE.74.020104.
관련도서
- Hirota, Ryogo, Atsushi Nagai, Jon Nimmo, and Claire Gilson. 2004. “Determinants and Pfaffians.” In The Direct Method in Soliton Theory. Cambridge Tracts in Mathematics. http://dx.doi.org/10.1017/CBO9780511543043.004.
- Barry M McCoy, Advanced Statistical Mechanics
- The Pfaffian solution of the Ising model DOI:10.1093/acprof:oso/9780199556632.003.0011
메타데이터
위키데이터
- ID : Q1189744
Spacy 패턴 목록
- [{'LEMMA': 'pfaffian'}]