SamplePublisher`GrassmannCalculus`
ToReducedFactoredForm  |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
The following is a 3-vector in 5-space with its expanded canonical 3-vector.
In[2]:=
 | ★ℬ |
5
e
1
e
2
e
3
e
4
e
5
e
1
e
2
e
3
e
4
e
5
e
1
e
2
e
3
e
5
| FastExteriorExpand |
Out[2]=
18⋀⋀+18⋀⋀-72⋀⋀+30⋀⋀-174⋀⋀-54⋀⋀+6⋀⋀-24⋀⋀+18⋀⋀
e
1
e
2
e
3
e
1
e
2
e
4
e
1
e
2
e
5
e
1
e
3
e
4
e
1
e
3
e
5
e
1
e
4
e
5
e
2
e
3
e
4
e
2
e
3
e
5
e
3
e
4
e
5
The following 'diagonalizes' the product on the basis vectors in ⋀⋀. The canonical m-vector is preserved.
e
1
e
2
e
3
In[3]:=
mProduct2=
[mProduct,⋀⋀]canonicalMVector2=
[mProduct2]canonicalMVector2canonicalMVector
| ToReducedFactoredForm |
e
1
e
2
e
3
| FastExteriorExpand |
Out[3]=
18+-⋀-+⋀(+-4)
e
1
e
4
3
4
e
5
3
e
2
5
e
4
3
29
e
5
3
e
3
e
4
e
5
Out[3]=
18⋀⋀+18⋀⋀-72⋀⋀+30⋀⋀-174⋀⋀-54⋀⋀+6⋀⋀-24⋀⋀+18⋀⋀
e
1
e
2
e
3
e
1
e
2
e
4
e
1
e
2
e
5
e
1
e
3
e
4
e
1
e
3
e
5
e
1
e
4
e
5
e
2
e
3
e
4
e
2
e
3
e
5
e
3
e
4
e
5
Out[3]=
True
Mapping on all the other 3-vectors:
GradeBasis[3]
In[4]:=
 | ToReducedFactoredForm |
 | GradeBasis |
Out[4]=
18 e 1 e 3 3 e 2 5 e 3 3 e 5 e 3 e 4 e 5 |
-72 e 1 e 3 3 e 2 29 e 3 12 3 e 4 4 e 3 4 e 4 4 e 5 |
30 e 1 e 2 5 3 e 5 5 3 e 2 5 e 3 9 e 5 5 3 e 2 5 e 4 29 e 5 5 |
-174 e 1 4 e 2 29 3 e 4 29 12 e 2 29 e 3 9 e 4 29 3 e 2 29 5 e 4 29 e 5 |
-54 e 1 e 3 3 4 e 2 3 29 e 3 9 e 4 e 2 3 5 e 3 9 e 5 |
6(5 e 1 e 2 e 5 e 1 e 3 e 1 e 4 e 5 |
-24 29 e 1 4 e 2 3 e 4 4 e 1 e 3 3 e 1 4 e 4 4 e 5 |
Undefined |
18(-3 e 1 e 3 29 e 1 3 4 e 2 3 e 4 5 e 1 3 e 2 3 e 5 |
The ⋀⋀ basis is because the associated columns in a matrix representation have a zero determinant.
e
2
e
4
e
5
Undefined
In[5]:=
(matrix=Coefficient[#,Basis]&/@(List@@mProduct))//MatrixForm(matrix245=matrix〚All,{2,4,5}〛)//MatrixFormDet[matrix245]
Out[5]//MatrixForm=
-190 | 28 | 96 | -14 | 140 |
57 | -9 | -30 | 4 | -43 |
-4 | 1 | 3 | 0 | 3 |
Out[5]//MatrixForm=
28 | -14 | 140 |
-9 | 4 | -43 |
1 | 0 | 3 |
Out[5]=
0
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