"Minors and plucker relations"의 두 판 사이의 차이
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imported>Pythagoras0 잔글 (찾아 바꾸기 – “<h5>” 문자열을 “==” 문자열로) |
imported>Pythagoras0 잔글 (찾아 바꾸기 – “</h5>” 문자열을 “==” 문자열로) |
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| 1번째 줄: | 1번째 줄: | ||
| − | ==introduction | + | ==introduction== |
# (mat = Array[Subscript[a, ##] &, {3, 3}]) // MatrixForm<br> Minors[mat] // MatrixForm<br> Minors[mat, 1] // MatrixForm<br> Minors[mat, 2] // MatrixForm<br> Minors[mat, 3] // MatrixForm | # (mat = Array[Subscript[a, ##] &, {3, 3}]) // MatrixForm<br> Minors[mat] // MatrixForm<br> Minors[mat, 1] // MatrixForm<br> Minors[mat, 2] // MatrixForm<br> Minors[mat, 3] // MatrixForm | ||
| 12번째 줄: | 12번째 줄: | ||
| − | ==3-term Plucker relation (Ptolemy relation) | + | ==3-term Plucker relation (Ptolemy relation)== |
* <math>\Delta _{i,k} \Delta _{j,l}=\Delta _{i,j} \Delta _{k,l}+\Delta _{i,l} \Delta _{j,k}</math> | * <math>\Delta _{i,k} \Delta _{j,l}=\Delta _{i,j} \Delta _{k,l}+\Delta _{i,l} \Delta _{j,k}</math> | ||
| 24번째 줄: | 24번째 줄: | ||
| − | ==Plucker relations | + | ==Plucker relations== |
* <math>\Delta _{1,2}\Delta _{12,13}=\Delta _{1,3}\Delta _{12,12}+\Delta _{1,1}\Delta _{12,23}</math> | * <math>\Delta _{1,2}\Delta _{12,13}=\Delta _{1,3}\Delta _{12,12}+\Delta _{1,1}\Delta _{12,23}</math> | ||
| 34번째 줄: | 34번째 줄: | ||
| − | ==Plucker coordinates of a Grassmannian | + | ==Plucker coordinates of a Grassmannian== |
| 40번째 줄: | 40번째 줄: | ||
| − | ==memo | + | ==memo== |
* [http://www.math.msu.edu/%7Emagyar/papers/MinorIdentities.pdf http://www.math.msu.edu/~magyar/papers/MinorIdentities.pdf] | * [http://www.math.msu.edu/%7Emagyar/papers/MinorIdentities.pdf http://www.math.msu.edu/~magyar/papers/MinorIdentities.pdf] | ||
* http://www.ams.org/journals/proc/2008-136-01/S0002-9939-07-09122-8/S0002-9939-07-09122-8.pdf | * http://www.ams.org/journals/proc/2008-136-01/S0002-9939-07-09122-8/S0002-9939-07-09122-8.pdf | ||
2012년 10월 28일 (일) 14:34 판
introduction
- (mat = Array[Subscript[a, ##] &, {3, 3}]) // MatrixForm
Minors[mat] // MatrixForm
Minors[mat, 1] // MatrixForm
Minors[mat, 2] // MatrixForm
Minors[mat, 3] // MatrixForm - Simplify[Subscript[a, 1,
3]*(-Subscript[a, 1, 2] Subscript[a, 2, 1] +
Subscript[a, 1, 1] Subscript[a, 2, 2]) +
Subscript[a, 1,
1]*(-Subscript[a, 1, 3] Subscript[a, 2, 2] +
Subscript[a, 1, 2] Subscript[a, 2, 3])]
3-term Plucker relation (Ptolemy relation)
- \(\Delta _{i,k} \Delta _{j,l}=\Delta _{i,j} \Delta _{k,l}+\Delta _{i,l} \Delta _{j,k}\)
- \(\Delta _{1,2}\Delta _{3,4}+\Delta _{1,4}\Delta _{2,3}=\Delta _{1,3}\Delta _{2,4}\)
- T := {{Subscript[a, 1, 1], Subscript[a, 1, 2], Subscript[a, 1, 3],
Subscript[a, 1, 4]}, {Subscript[a, 2, 1], Subscript[a, 2, 2],
Subscript[a, 2, 3], Subscript[a, 2, 4]}}
Minor[i_, j_] := Det[{Transpose[T]i, Transpose[T]j}]
Minor[1, 2] - Simplify[Minor[1, 2] Minor[3, 4] + Minor[1, 4] Minor[2, 3]]
Simplify[Minor[1, 3] Minor[2, 4]]
Plucker relations
- \(\Delta _{1,2}\Delta _{12,13}=\Delta _{1,3}\Delta _{12,12}+\Delta _{1,1}\Delta _{12,23}\)
- \Delta _{12,12}\text{:=}-a_{1,2} a_{2,1}+a_{1,1} a_{2,2}\Delta _{12,23}\text{:=}-a_{1,3} a_{2,2}+a_{1,2} a_{2,3}\Delta _{1,3}\text{:=}a_{1,1}\Delta _{1,3}\text{:=}a_{1,3}\Delta _{1,3}\Delta _{12,12}+\Delta _{1,1}\Delta _{12,23}
Plucker coordinates of a Grassmannian