Conformal Geometric Algebra in Stochastic Optimization Problems ...

2 CHAPTER 1. INTRODUCTION
with a simple, i.e. low dimensional, geometric algebra (indicated in brackets). Each
of the quoted constructs possesses its own specific notation even though all of them
can be treated in a unified manner by means of GA. A similar situation holds for
the 3D-space with its scalar product and the vector cross product: in GA a single
operation encompasses both of the ‘constructs’. These then arise naturally insofar
as they reflect the symmetric and anti-symmetric part, respectively, of this only
product. And it does so irrespective of the dimension, i.e. it generalizes to algebras
built on spaces with dimension different from three. The portrayed product is of
course the algebra product - the geometric product. By the excess of information,
it follows that, not least from a sheerly geometrical point of view, the geometric
product must be reversible if one of its operands is still known. It is hence the
accomplishment of the geometric product that virtually all elements of the algebra
have a multiplicative inverse. As the algebra always comprises the vector space it
is built on, it can in particular be ‘divided’ by a vector.
Clearly, every task may likewise be accomplished, in a way, by means of the standard
vector algebra. It may be permitted to say: ‘this is comparable to a pipe fitter
who incessantly utilizes hammer and screwdriver, although, for example, the pipe
wrench in his toolkit beside him would be much more practicable.’ Since GA can
be regarded as a universal framework, problems should be modeled with it from the
beginning on. It is then still possible to make the transition to a suitable matrix
representation if a numerical evaluation is due. This is exactly the way chosen in
this work - no modeling option is abandoned before the time.
1.1.1 GA - General Things
The term �p,q denotes the geometric algebra over an n-dimensional vector space
�n , n = p + q, equipped with signature (p,q) ∈ �2 . The latter will, however,
become of importance not before the next chapter. The dimension of the algebra
is then 2n . Its elements are termed multivectors, where it is fully sufficient to treat
a multivector as a vector from � 2n
as long as its respective algebra is known. The
components of a multivector comprise n+1 different grades, each with a multiplicity
according to the n + 1th row of Pascal’s triangle. By convention, the components
of a multivector are ordered according to their grade: one scalar (grade zero), n
vectors (grade one) and the higher grades. Note that this one-to-one corresponds
to the quadruplet representation of the quaternions with basis 1 = (1,0,0,0), i =
(0, 1, 0, 0), j = (0,0,1,0) and k = (0,0,0,1). Hence, imposing the common order
1,i, j,k, multivector k, for instance, correctly reflects the element of grade two.
Moreover, observe that just like for the complex numbers, e.g. z = a + ib, a
number a can be added to a vector ib in a meaningful way. The simple relationships
can be looked up in many textbooks. Important for this work is as well that for
every GA there exists at least one matrix representation with real square matrices
representing multivectors. Building the geometric product simply amounts to the
matrix product. An illustrative example are the Pauli matrices (of size 4 × 4 in
the real case): the identity matrix holds the scalar component. The Pauli matrices
themselves can be identified with the three vector valued basis elements of the
eight-dimensional Pauli algebra. The existence of the isomorphism between GA and