"Minors and plucker relations"의 두 판 사이의 차이
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| 3번째 줄: | 3번째 줄: | ||
| − | < | + | <h5>introduction</h5> |
# (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 | ||
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| + | <h5>Ptolemy relation</h5> | ||
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| + | <math>\Delta _{1,2}\Delta _{3,4}+\Delta _{1,4}\Delta _{2,3}=\Delta _{1,3}\Delta _{2,4}</math> | ||
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| + | # T := {{Subscript[a, 1, 1], Subscript[a, 1, 2], Subscript[a, 1, 3],<br> Subscript[a, 1, 4]}, {Subscript[a, 2, 1], Subscript[a, 2, 2],<br> Subscript[a, 2, 3], Subscript[a, 2, 4]}}<br> Minor[i_, j_] := Det[{Transpose[T][[i]], Transpose[T][[j]]}]<br> Minor[1, 2] | ||
| + | # Simplify[Minor[1, 2] Minor[3, 4] + Minor[1, 4] Minor[2, 3]]<br> Simplify[Minor[1, 3] Minor[2, 4]] | ||
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| + | <h5>Plucker relations</h5> | ||
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| + | <math>\Delta _{1,2}\Delta _{12,13}=\Delta _{1,3}\Delta _{12,12}+\Delta _{1,1}\Delta _{12,23}</math> | ||
# \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} | # \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} | ||
2011년 3월 9일 (수) 06:20 판
http://www.math.msu.edu/~magyar/papers/MinorIdentities.pdf
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])]
Ptolemy relation
\(\Delta _{1,2}\Delta _{3,4}+\Delta _{1,4}\Delta _{2,3}=\Delta _{1,3}\Delta _{2,4}\)
- 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}