1. A Mathematica Program for Geometric Algebra
Gordon Erlebacher Garret Sobczyk

2. Mi vida a Udla Learn Spanish Learn to dance Learn about life at Udla
Work with Garret
Not necessarily in the above order!!

3. Hard at work

4. Learning spanish a la Fiesta y a la Iglesia

5. Learning spanish while dancing

6. My influence on Garret

7. Gordon repents for his fun

8. Gordon starts serious work

9. Objectives
Develop a new algebra of 2x2 matrices whose elements are in the geometric algebra G3
Develop a Mathematica program to perform symbolic calculations in this algebra
Reduce calculation errors
Check calculations done manually
Try out new ideas
Peforms complex computations

10. Contents
Algebras
Mathematica Program
Application

11. What is an Algebra?
Combination of a Field of scalars , a vector space (+) and a ring (x)
Commutativity of elements of the Field and elements of the algebra

12. Examples of Algebras All non-invertible matrices Unitary matrices
Integers
Real numbers
Set of polynomials with integer coefficients

13. Concepts from Geometric Algebra
Let (vector space)
Graded Algebra:
Non-commutative
Geometric interpretations
are vectors
are scalars
are bivectors (tangent plane)
are trivectrors (volume elements)
Scalar product
Outer product

14. Geometric interpretation 2D vector space
Scalar  point (grade 0)
Directed line  vector (grade 1)
Directed plane  bivector (grade 2)
Directed volume  trivector (grade 3)

15. 2D: G2 Basis : 4=22 elements one scalar: 1 two vectors:
one pseudoscalar, bivector:

16. 3D: G3 Basis : 8=23 elements one scalar: 1 three orthonormal vectors:
three bivectors:
one pseudoscalar, bivector:

17. Generalizations
Geometric algebra generalizes nicely to higher dimensions n-D

18. Element of G3 An element of takes the form
has properties of imaginary number, i.e.,
for all
Rewrite element g as
4 complex numbers encode the same information as the 8 elements of G3 1
2,3,4

19. Paravector
Element g of G3 is written as
called a paravector

20. Operations on G3 Geometric product
Rewriting in terms of complex coefficients, we find that
/\ is associative, is not
is symmetric part of
is the antisymmetric part of

21. Algebra: G41 algebra The algebra is generated by the vector space V41
Signature of V41 is (++++-), or
Pseudoscalar: 5-D volume element
32 basis functions:
1 scalar, 5 vectors, 10 bivectors (2-vector), 10 trivectors, 5 4-vectors, 1 pseudoscalar (5-vector)

22. Isomorphism between Gn and matrix algebra
It can be shown that every G2n (in ) is isomorphic to the algebra Mm of mxm matrices of reals (in ) .
Thus, if then (homomorphism)

23. Isomorphism G41 M2(G3) Element g of G41 is isomorphic to
Element a of G3 is isomorphic to

24. Degrees of freedom G4,1 has 25=32 degrees of freedom
A 2x2 matrix has 4 degrees of freedom
G3 has 8 degrees of freedom
Therefore, M2(G3) has 32 degrees of freedom, consistent with G41

25. Why M2(G3)?
Combine the advantages of matrix algebra with that of Geometric (Clifford) algebras
2x2 matrices are small and simple to manipulate
Extensive literature on matrices
G3 is closely related to the standard vector algebra of Gibbs

26. Why G41?
5-dimensional vector space is a superset of the following useful spaces:
Euclidean space
Quaternions
Affine space
Projective space
Horosphere
Space of special relativity

27. Mathematica Parent company: Wolfram
Powerful software package for symbolic manipulation
Exists for more than 15 years
Main competitor: Maple (with similar capabilities, but a different programming style)

28. Paravector

29. ParaMatrix

30. Operations in G41
The display is independent of internal representation elements

31. Expansions Display depends on internal representation
Choose the representation that is most convenient
pV[a] operates the fastest but it is not possible to work with scalar and vector components

32. More complex operations
Expansion
Simplification
scalarPart, vectorPart
Determinant
Characteristic polynomial
Conversion routines between G41 and M2(G3)
more …

33. Expansion expandAll[x_] := FixedPoint[expandAllOnce,x]
expandAllOnce[x_] := Module[{y=x},
(* Conjugation should probably be done near the
y = y //. flattenGeomRules;
y = y //. geomRules;
y = y //. expandGeomRules;
y = y //. expandpVRules;
y = y //. expandDotRules;
y = y //. expandWedgeRules;
y = y //. conjugationRules;
y = y //. inversionRules;
y = y //. reversionRules;
y= y //. tripleRules; (* Need a display for
y= y //. orderRules;
y = y // ExpandAll; y ]

34. Matrix multiplication

35. Expansion

36. Futher expansion

37. Techniques for simplification
Identify scalar components in products and extract them
Isoloate scalar components using scal[] to avoid problems in complex expressions Flatten out geometric products to take associativity into account

40. Checking identities
Checking through use of random numbers (not demonstrated here)

41. We expect that o1 = o2

42. = 0 ???
Yes!!!
Convert to 4-component form

43. Determinant of g in M2(G3)
Consider a 2x2 matrix
Element of G3: g = g0+gi ei has the matrix representation
The determinant of M is simply the determinant of the 4x4 matrix

44. Goal
Compute determinant using only matrix elements and their conjugates
Use Gauss elimination, taking non-conjugation into account
Reminder, conjugation of is

45. Determinant of
The determinant is simply (in )

46. Objective Express Det[m44] using only and their conjugates
How to use Mathematica to do this automatically?.

47. Determinant Properties
Multiplication by a vector of G3
Linear combinations of row (or columns):

48. Gauss elimination
Remember: elements of G3 are not commutative

49. Gauss Algorithm
This form led to a Gauss-like algorithm with the result which can be rewritten as

50. General Inverse

51. How about . Try Gauss elimination Leads to very complex formulas
Pivot[1,1]
Leads to very complex formulas

52. Using Mathematica

53. Simplify further

54. Final answer
But …

55. How about
So far, we are not able to find a formula for the determinant (by hand or with Mathematica!)
This is our next goal
Try to find a recursive formula to compute as a function of

56. First step
Understand better the properties of as a function of lower order traces. Mathematica is required for this.

57. Examples Tr (a,b) = <a,b>0=scalarPart[oGeom[ab]] Tr (a,b,c)
Tr (a,b,c,d)
Tr(a,b,c,d,e)
Tr (a,b,c,d,e,f)
Higher order …

58. Possible Approach Define
Find formulas for and as a function of and where m < n
So far we have been unsuccessful. Formulas get complicated very fast

59. First few traces
<ab>0
<abc>0
<abcd>0
<abcde>0

60. First few vector parts <ab>1 <abc>1 <abcd>1
<abcde>1

61. Conclusions Potentially powerful new algebra
Subalgebras include: Euclidean space, affine space, projective space, horosphere, relativity, twistors, quaternions, etc.
Powerful Mathematica program available that operates similarly to pen and paper
This work has many potential extensions
Search for more general formulas is underway

62. Muchas Gracias por todo!!!
Preguntas???