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A Mathematica Program for Geometric Algebra

- Gordon ErlebacherGarret Sobczyk

Mi vida a Udla

- Learn Spanish
- Learn to dance
- Learn about life at Udla
- Work with Garret

Not necessarily in the above order!!

Hard at work

Learning spanish a la Fiesta y a la Iglesia

Learning spanish while dancing

My influence on Garret

Gordon repents for his fun

Gordon starts serious work

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

Contents

- Algebras
- Mathematica Program
- Application

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

Examples of Algebras

- All non-invertible matrices
- Unitary matrices
- Integers
- Real numbers
- Set of polynomials with integer coefficients

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

Geometric interpretation2D vector space

- Scalar ?? point (grade 0)
- Directed line ?? vector (grade 1)
- Directed plane ?? bivector (grade 2)
- Directed volume ?? trivector(grade 3)

2D G2

- Basis 422 elements
- one scalar 1
- two vectors
- one pseudoscalar, bivector

3D G3

- Basis 823 elements
- one scalar 1
- three orthonormal vectors
- three bivectors
- one pseudoscalar, bivector

Generalizations

- Geometric algebra generalizes nicely to higher

dimensions n-D

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

Paravector

- Element g of G3 is written as
- called a paravector

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

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)

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)

Isomorphism G41 M2(G3)

- Element g of G41 is isomorphic to
- Element a of G3 is isomorphic to

Degrees of freedom

- G4,1 has 2532 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

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

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

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)

Paravector

ParaMatrix

Operations in G41

The display is independent of internal

representation elements

Expansions

- Display depends on internal representation

Choose the representation that is most

convenient pVa operates the fastest but it is

not possible to work with scalar and vector

components

More complex operations

- Expansion
- Simplification
- scalarPart, vectorPart
- Determinant
- Characteristic polynomial
- Conversion routines between G41 and M2(G3)
- more

Expansion

- expandAllx_ FixedPointexpandAllOnce,x
- expandAllOncex_ Moduleyx,
- ( 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

Matrix multiplication

Expansion

Futher expansion

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

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Checking identities

- Checking through use of random numbers (not

demonstrated here)

We expect that o1 o2

Convert to 4-component form

Determinant of g in M2(G3)

- Consider a 2x2 matrix
- Element of G3 g g0gi ei has the matrix

representation - The determinant of M is simply the determinant

of the 4x4 matrix

Goal

- Compute determinant using only matrix elements

and their conjugates - Use Gauss elimination, taking non-conjugation

into account - Reminder, conjugation of is

Determinant of .

- The determinant is simply (in )

Objective

- Express Detm44 using onlyand their conjugates
- How to use Mathematica to do this automatically?.

Determinant Properties

- Multiplication by a vector of G3
- Linear combinations of row (or columns)

Gauss elimination

Remember elements of G3 are not commutative

Gauss Algorithm

- This form led to a Gauss-like algorithm with the

resultwhich can be rewritten as

General Inverse

How about .

- Try Gauss elimination

Pivot1,1

Leads to very complex formulas

Using Mathematica

Simplify further

Final answer

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

First step

- Understand better the properties ofas a

function of lower order traces. Mathematica is

required for this.

Examples

- Tr (a,b) lta,bgt0scalarPartoGeomab
- Tr (a,b,c)
- Tr (a,b,c,d)
- Tr(a,b,c,d,e)
- Tr (a,b,c,d,e,f)
- Higher order

Possible Approach

- Define
- Find formulas for and as a

function of and where m lt n - So far we have been unsuccessful. Formulas get

complicated very fast

First few traces

ltabgt0

ltabcgt0

ltabcdgt0

ltabcdegt0

First few vector parts

ltabgt1

ltabcgt1

ltabcdgt1

ltabcdegt1

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

- Muchas Gracias por todo!!!
- Preguntas???

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