"스미스 표준형 (Smith normal form)"의 두 판 사이의 차이
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==관련논문== | ==관련논문== | ||
| + | * Shah, Syed Waqar Ali. “Smith Normal Form of Matrices Associated with Differential Posets.” arXiv:1510.00588 [math], October 2, 2015. http://arxiv.org/abs/1510.00588. | ||
* Dumas, Jean-Guillaume, Frank Heckenbach, David Saunders, and Volkmar Welker. 2003. “Computing Simplicial Homology Based on Efficient Smith Normal Form Algorithms.” In Algebra, Geometry and Software Systems, edited by Michael Joswig and Nobuki Takayama, 177–206. Springer Berlin Heidelberg. http://link.springer.com/chapter/10.1007/978-3-662-05148-1_10. | * Dumas, Jean-Guillaume, Frank Heckenbach, David Saunders, and Volkmar Welker. 2003. “Computing Simplicial Homology Based on Efficient Smith Normal Form Algorithms.” In Algebra, Geometry and Software Systems, edited by Michael Joswig and Nobuki Takayama, 177–206. Springer Berlin Heidelberg. http://link.springer.com/chapter/10.1007/978-3-662-05148-1_10. | ||
2015년 10월 4일 (일) 19:09 판
개요
- 정수환 위에서의 선형대수학
- 정수 행렬에 대한 표준형의 하나
- 호몰로지, 유한생성 아벨군의 기본정리 등에서 활용
예
- 정수 계수 행렬
$$ \left( \begin{array}{ccccc} 1 & -5 & 0 & 10 & -15 \\ 0 & 4 & 0 & -8 & 12 \\ 3 & -3 & -2 & 6 & -9 \\ 1 & -1 & 0 & 2 & -3 \\ \end{array} \right) $$
- 이 행렬의 스미스 표준형 (Smith normal form)은 다음과 같다
$$ \left( \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 4 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} \right) =\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 3 & 3 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ -1 & -1 & 0 & 1 \end{array} \right).\left( \begin{array}{ccccc} 1 & -5 & 0 & 10 & -15 \\ 0 & 4 & 0 & -8 & 12 \\ 3 & -3 & -2 & 6 & -9 \\ 1 & -1 & 0 & 2 & -3 \end{array} \right).\left( \begin{array}{ccccc} 1 & 0 & 5 & 0 & 0 \\ 0 & 0 & 1 & 2 & -3 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array} \right) $$
관련된 항목들
매스매티카 파일 및 계산 리소스
- https://docs.google.com/file/d/0B8XXo8Tve1cxb1ZzczQzRVBiemc/edit
- Smith Normal Forms Wolfram library archive
관련도서
- Norman, Christopher. 2012. Finitely Generated Abelian Groups and Similarity of Matrices over a Field, Springer.
관련논문
- Shah, Syed Waqar Ali. “Smith Normal Form of Matrices Associated with Differential Posets.” arXiv:1510.00588 [math], October 2, 2015. http://arxiv.org/abs/1510.00588.
- Dumas, Jean-Guillaume, Frank Heckenbach, David Saunders, and Volkmar Welker. 2003. “Computing Simplicial Homology Based on Efficient Smith Normal Form Algorithms.” In Algebra, Geometry and Software Systems, edited by Michael Joswig and Nobuki Takayama, 177–206. Springer Berlin Heidelberg. http://link.springer.com/chapter/10.1007/978-3-662-05148-1_10.