GrassmannCalculus`
ToCommonFactor  |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
Here is the common factor of two 2-elements in a 3-space. Geometrically, this may be interpreted as the common vector of two bivectors, or the point of intersection between two lines in a plane.
In[2]:=
★A;
[
 | ToCommonFactor |
(
x
⋀y)⋁(u⋀v)]Out[2]=
★c(y〈u⋀v⋀x〉-x〈u⋀v⋀y〉)
If the first factor is a simple exterior product of 1-elements (for example x⋀y as in the above example), the expansion is carried out by decomposing this first factor. The arbitrary scalar factor is called the congruence factor. The expression (using ) is interpreted as the scalar coefficient of the n-element when expressed in terms of the current basis.
★c
〈u⋀v⋀x〉
AngleBracket
u⋀v⋀x
You can do common factor calculations with arbitrary Grassmann expressions.
In[3]:=
 | ★ℬ |
3
| ToCommonFactor |
α
2
β
2
γ
2
Out[3]=
1+2★cu⋁⋀-2★cu⋀⋁
β
2
γ
2
α
2
α
2
β
2
γ
2
You can also do common factor calculations when the elements are expressed in terms of a basis. Here is the regressive product of two bivectors in a 3-space.
In[4]:=
★A;X=
[
 | ComposeBasisForm |
(
x
⋀y)⋁(u⋀v)]Out[4]=
(++)⋀(++)⋁(++)⋀(++)
e
1
x
1
e
2
x
2
e
3
x
3
e
1
y
1
e
2
y
2
e
3
y
3
e
1
u
1
e
2
u
2
e
3
u
3
e
1
v
1
e
2
v
2
e
3
v
3
In[5]:=
| ToCommonFactor |
Out[5]=
★c(((-)+(-+)+((-+)+(-)))+((-)+(-)+((-+)+(-+)))+(((-+)+(-))+((-)+(-+))))
e
1
u
3
v
1
x
2
y
1
x
1
y
2
u
2
v
1
x
3
y
1
x
1
y
3
u
1
v
3
x
2
y
1
x
1
y
2
v
2
x
3
y
1
x
1
y
3
e
2
u
3
v
2
x
2
y
1
x
1
y
2
u
1
v
2
x
3
y
2
x
2
y
3
u
2
v
3
x
2
y
1
x
1
y
2
v
1
x
3
y
2
x
2
y
3
e
3
v
3
u
2
x
3
y
1
x
1
y
3
u
1
x
3
y
2
x
2
y
3
u
3
v
2
x
3
y
1
x
1
y
3
v
1
x
3
y
2
x
2
y
3
ToCommonFactor can calculate the common factor possessed by any number of elements. Here we compute an expression for the common 1-element between three 3-elements in a 4-space.
In[6]:=
| ★ℬ |
4
| ToCommonFactor |
(
x
⋀y⋀z)⋁(u⋀v⋀w)⋁(p⋀q⋀r)]Out[6]=
2
★c
The result becomes a little more apparent if we expand and simplify it to collect the scalar coefficients of the 1-elements x, y, and z.
In[7]:=
| ★ |
Out[7]=
2
★c
2
★c
2
★c
Here the congruence factor ★c appears squared. This is a consequence of the fact that the regressive product has been applied twice.
ToCommonFactor can of course also perform its calculations with specific numerically defined elements. Here we generate the regressive product of four random 4-elements in a 5-space.
In[8]:=
| ★ℬ |
5
| ComposeBasisForm |
a
4
a
4
a
4
a
4
a
_,_
Out[8]=
(0.29338⋀⋀⋀+0.588126⋀⋀⋀+0.425378⋀⋀⋀+0.994102⋀⋀⋀+0.212986⋀⋀⋀)⋁(0.317637⋀⋀⋀+0.207955⋀⋀⋀+0.275859⋀⋀⋀+0.707407⋀⋀⋀+0.504748⋀⋀⋀)⋁(0.728213⋀⋀⋀+0.902312⋀⋀⋀+0.599313⋀⋀⋀+0.363975⋀⋀⋀+0.452954⋀⋀⋀)⋁(0.67403⋀⋀⋀+0.546525⋀⋀⋀+0.651856⋀⋀⋀+0.109399⋀⋀⋀+0.466954⋀⋀⋀)
e
1
e
2
e
3
e
4
e
1
e
2
e
3
e
5
e
1
e
2
e
4
e
5
e
1
e
3
e
4
e
5
e
2
e
3
e
4
e
5
e
1
e
2
e
3
e
4
e
1
e
2
e
3
e
5
e
1
e
2
e
4
e
5
e
1
e
3
e
4
e
5
e
2
e
3
e
4
e
5
e
1
e
2
e
3
e
4
e
1
e
2
e
3
e
5
e
1
e
2
e
4
e
5
e
1
e
3
e
4
e
5
e
2
e
3
e
4
e
5
e
1
e
2
e
3
e
4
e
1
e
2
e
3
e
5
e
1
e
2
e
4
e
5
e
1
e
3
e
4
e
5
e
2
e
3
e
4
e
5
Applying to this product gives the common 1-element. This takes a minute.
ToCommonFactor
In[9]:=
Xc=
[X]
| ToCommonFactor |
Out[9]=
3
★c
e
1
e
2
e
3
e
4
e
5
We can easily verify that this 1-element is common to each of the four 4-elements by tabulating their exterior products, and comparing the results to zero.
In[10]:=
Y=Table
[Xc⋀X〚i〛]//Chop,{i,1,4}
 | ★ |
Out[10]=
{0,0,0,0}
In[11]:=
Clear[X,Xc,Y]
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