Conformal Geometric Algebra in Stochastic Optimization Problems ...

246 APPENDIX A. SELECTED ASPECTS UNDERLYING THIS WORK
is the most general form the outer product of k vectors a1, a2, ..., ak may take on.
A condition, or rather the definition for this general case is that each summand of
equation (2.27) has to occur. It is therefore assumed that θσ �= 0 for all σ ∈ S(k).
Here it is shown that the coefficients θσ must then all be equal in their absolute
value, that is |θσ| = const.
Consider two arbitrary terms of the sum in equation (2.27)
fσ = θσ a σ(1)a σ(2) ...a σ(k) and f˜σ = θ˜σ a ˜σ(1)a ˜σ(2) ...a ˜σ(k).
It exists a permutation π ∈ S(k) so that π ◦ σ = ˜σ or ˜σ −1 = σ −1 ◦ π −1 ,
equivalently. Note that with the concatenation ω := ˜σ −1 ◦σ, ω ∈ S(k), and so with
ω = σ −1 ◦ π −1 ◦ σ, it is ˜σ ◦ ω = σ. Hence permuting the vectors a1,a2,...,ak
with ω beforehand yields
f˜σ(a ω(1), a ω(2),...,a ω(k)) = θ˜σ a σ(1)a σ(2) ...a σ(k)
= θ˜σ fσ(a1, a2,...,ak)/θσ .
But f is supposed to be alternating which, on the other hand, implies
f(a ω(1), a ω(2), ...,a ω(k)) = sgn(ω)f(a1, a2, ...,ak) .
Thus given two arbitrary permutations σ and ˜σ, a permutation ω of the argument
vectors of f can be found in such a way that the summand fσ plays the role of f˜σ,
and vice versa. Consequently, for the quotient θ˜σ/θσ it is required that 2
θ˜σ
θσ
= sgn(ω) = sgn(σ −1 )sgn(π −1 )sgn(σ) = sgn(π) (A.13)
since sgn(σ −1 )sgn(σ) = 1 and sgn(π −1 ) = sgn(π). This already shows that the absolute
values |θσ| of the summands in equation (2.27) must be identical. Moreover,
from π ◦ σ = ˜σ it can be deduced that
sgn(π ◦ σ) = sgn(˜σ) ⇐⇒ sgn(π) = sgn(˜σ)
sgn(σ) .
In conjunction with equation (A.13) it may eventually be defined
θσ := c sgn(σ), c > 0, (A.14)
where c = const denotes a positive scalar from the reals �.
2 The sgn function is a group homomorphism. It therefore holds that
sgn(σ1 ◦ σ2) = sgn(σ1) sgn(σ2) σ1, σ2 ∈ S .