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Community-contributed installable additions to the Wolfram Language
Matrix Gateway
Interface between multivectors and matrices.
MultivectorMatrix | |
MatrixMultivector | |
MultivectorBlock |
by Garret Sobczyk, 2019.
Let’s work with 5-dimensional geometric algebras and
3,0
2,1
In[6]:=
G1=
[3,0]G2=
[2,1]
 | GeometricAlgebra |
| GeometricAlgebra |
Out[6]=
3
Out[7]=
2,1
And generate random numerical multivectors in these algebras
In[8]:=
SeedRandom[0];v=
[Rationalize[RandomReal[{-1,1},8],0.01],G1]w=
[Rationalize[RandomReal[{-1,1},8],0.01],G2]
| Multivector |
| Multivector |
Out[9]=
3
10
3
11
e
1
3
8
e
2
1
8
e
3
7
8
e
12
17
18
e
13
9
17
e
23
3
11
e
123
Out[10]=
-++-++++-+
4
5
2
7
e
1
13
19
e
2
5
8
e
1
4
5
e
12
2
7
e
1
1
5
13
e
2
1
12
23
e
12
1
Any multivector can be transformed into a matrix (square MultivectorArray of rank 2) containing lower dimensional multivectors as its components
In[8]:=
MultivectorMatrix[v]
Out[8]=
| ||
|
In[9]:=
MultivectorMatrix[w]
Out[9]=
| ||
|
Notice that matrix components of contain multivectors of , and components of contain which are lowest dimensional algebras with the same sign of pseudoscalar squared, -1 and 1 respectivelyMatrix components of also contain the imaginary units , which come from the conversion to a balanced algebra ≡(,,)(,,)≡.
3,0
0,1
2,1
1,0
3,0
3,0
e
1
e
2
e
3
e
1
e
2
e
3
2,1
Converting from one algebra to another is easy:
In[10]:=
 | ConvertGeometricAlgebra |
Out[10]=
-++-+-++-++-
4
5
2
7
e
1
13
19
e
2
5
8
e
3
4
5
e
12
2
7
e
13
5
13
e
23
12
23
e
123
In[11]:=
| ConvertGeometricAlgebra |
Out[11]=
3
10
3
11
e
1
3
8
e
2
8
e
1
7
8
e
12
17
18
e
1
1
9
17
e
2
1
3
11
e
12
1
In[12]:=
| ConvertGeometricAlgebra |
 | ConvertGeometricAlgebra |
Out[12]=
True
In[13]:=
| ConvertGeometricAlgebra |
| ConvertGeometricAlgebra |
Out[13]=
True
The inverse operation of MultivectorMatrix is MatrixMultivector, with an optional additional argument of desired geometric algebra, because there are exist different geometric algebra signatures with dimension 3:
3+1
In[14]:=
MatrixMultivector[MultivectorMatrix[v],G1]vMatrixMultivector[MultivectorMatrix[w],G2]w
Out[14]=
True
Out[15]=
True
Multiplying matrices is equivalent to multiplying multivectors (make sure to multiply within one algebra to avoid coercion):
In[16]:=
MatrixMultivectorMultivectorMatrix[v]**MultivectorMatrix
[w,G1],G1===v**
[w,G1]MatrixMultivectorMultivectorMatrix
[v,G2]**MultivectorMatrix[w],G2===
[v,G2]**w
 | ConvertGeometricAlgebra |
 | ConvertGeometricAlgebra |
| ConvertGeometricAlgebra |
| ConvertGeometricAlgebra |
Out[16]=
True
Out[17]=
True
Using this Matrix Gateway, any conventional operation on matrices () can be imported into geometric algebra.
The computation is based on the following simple realization. Given a 1-dimensional algebra with pseudoscalar that squares to -1:
In[18]:=
=I;
In[19]:=
^2
Out[19]=
-1
Applying any matrix function (like ) to a matrix with scalar and pseudoscalar components and is equivalent to applying it to the matrix and its conjugate and then adding them together (with appropriate coefficients):
MatrixExp
X
Y
In[20]:=
X=RandomReal[1,{4,4}];Y=RandomReal[1,{4,4}];M=X+Y;
In[23]:=
a=MatrixExp[X+Y];b=MatrixExp[X-Y];+(-)-MatrixExp[M]//Chop
a+b
2
a-b
2
Out[25]=
{{0,0,0,0},{0,0,0,0},{0,0,0,0},{0,0,0,0}}
Where coefficient was set exactly for the expression to simplify as +=a, which makes this completely redundant, because if there is a built-in way to do exponentiation on complex matrices, we can just do it directly. But for hyperbolic case with =1 and treating itself as gives the correct procedure:
-
a+b
2
a-b
2
2
1
In[26]:=
=
[1]["Pseudoscalar"];
| GeometricAlgebra |
In[27]:=
^2
Out[27]=
1
In[28]:=
a=MatrixExp[X+Y];b=MatrixExp[X-Y];MultivectorArray+MultivectorFunction[Exp,MultivectorArray[X+Y]]
a+b
2
a-b
2
Out[30]=
True
For example computing an inverse multivector uses matrix inversion function:
In[31]:=
Inverse[v]MatrixMultivector[MultivectorFunction[1/#&,MultivectorMatrix[v]],G1]
Out[31]=
True
In[32]:=
Inverse[v]**v//N//ChopInverse[w]**w//N//Chop
Out[32]=
1.
Out[33]=
1.
Or exponential and logarithm:
In[34]:=
MultivectorFunction[Exp,w]//NMultivectorFunction[Log,v]//N
Out[34]=
0.615653+0.0261969+(-0.467507)+0.0711741+0.224935+0.384702+(-0.113095)+0.517206
e
1
e
2
e
1
e
12
e
1
1
e
2
1
e
12
1
Out[35]=
0.337383+0.+(0.089309+1.94289×)+(0.057115+0.)+(0.380016+3.60822×)+(0.763926-3.33067×)+(0.979259+0.)+(-0.567895-8.32667×)+(0.250175+0.)
-16
10
e
1
e
2
-16
10
e
3
-16
10
e
12
e
13
-16
10
e
23
e
123
Check and identities:
Log[Exp[v]]==v
Exp[Log[v]]==v
In[6]:=
MultivectorFunction[Log,MultivectorFunction[Exp,v]]v//FullSimplify
Out[6]=
True
Nest multivector blocks: