A Mathematica Program for Geometric Algebra

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A Mathematica Program for Geometric Algebra

• Gordon ErlebacherGarret Sobczyk

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Mi vida a Udla

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

Not necessarily in the above order!!
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Hard at work
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Learning spanish a la Fiesta y a la Iglesia
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Learning spanish while dancing
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My influence on Garret
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Gordon repents for his fun
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Gordon starts serious work
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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

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Contents

• Algebras
• Mathematica Program
• Application

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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

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Examples of Algebras

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

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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
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Geometric interpretation2D vector space

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

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2D G2

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

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3D G3

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

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Generalizations

• Geometric algebra generalizes nicely to higher
  dimensions n-D

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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

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2,3,4
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Paravector

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

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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

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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)

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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)

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Isomorphism G41 M2(G3)

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

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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

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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

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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

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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)

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Paravector
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ParaMatrix
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Operations in G41
The display is independent of internal
representation elements
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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
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More complex operations

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

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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

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Matrix multiplication
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Expansion
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Futher expansion
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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)

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We expect that o1 o2
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Convert to 4-component form
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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

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Goal

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

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Determinant of .

• The determinant is simply (in )

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Objective

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

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Determinant Properties

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

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Gauss elimination
Remember elements of G3 are not commutative
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Gauss Algorithm

• This form led to a Gauss-like algorithm with the
  resultwhich can be rewritten as

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General Inverse
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How about .

• Try Gauss elimination

Pivot1,1
Leads to very complex formulas
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Using Mathematica
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Simplify further
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Final answer
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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

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First step

• Understand better the properties ofas a
  function of lower order traces. Mathematica is
  required for this.

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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

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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

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First few traces
ltabgt0
ltabcgt0
ltabcdgt0
ltabcdegt0
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First few vector parts
ltabgt1
ltabcgt1
ltabcdgt1
ltabcdegt1
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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

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• Muchas Gracias por todo!!!
• Preguntas???