Conformal Geometric Algebra in Stochastic Optimization Problems ...

34 CHAPTER 2. GEOMETRIC ALGEBRA
In carrying over the previous result to the case f(ev(1),ev(2),...,e v(k)) one obtains
f(ev(1),ev(2),...,e v(k)) = c �
σ∈S(k)
= c �
=
⎛
σ∈S(k)
⎝c �
σ∈S(k)
sgn(σ)e v(σ(1))e v(σ(2)) ...e v(σ(k))
�
�
sgn(σ) sgn(σ)ev(1)ev(2) ...e v(k)
sgn(σ) 2
⎞
⎠ e v(1)e v(2) ...e v(k)
= � ck! � e v(1)e v(2) ...e v(k) , c > 0. (2.29)
Notice that e v(1)e v(2) ...e v(k) represents an ordered basis blade due to the definition
(2.25) of v. The expression for f ultimately becomes
f(a1, a2,...,ak) = � ck! � �
v∈I k/n
det(A|v) e v(1)e v(2) ...e v(k) , c > 0.
Consequently, the outer product is uniquely defined up to a positive constant c ∈ �.
Regarding the outer product of basis vectors in equation (2.29) it is sensible to set
the factor c to c := 1/k!. The function f can now be replaced with the fully
determined outer product, symbolized by � . Let
� (a1,a2,...,ak) := a1 ∧ a2 ∧ ... ∧ ak =:
k�
aj .
Then the definitions for the outer product of k vectors a1, a2, ..., ak are
� (a1, a2,...,ak) := 1
k!
and � (a1, a2,...,ak) := �
with
�
σ∈S(k)
v∈I k/n
j=1
sgn(σ)a σ(1)a σ(2) ...a σ(k)
(2.30)
det(A|v) e v(1)e v(2) ...e v(k) , (2.31)
I k/n := {(v1,v2,...,vk) | 1 ≤ v1 < v2 < . .. < vk ≤ n} .
Note that the application of equation (2.30) to the outer product of mutually different
basis vectors, say ei1 ∧ei2 ∧... ∧eir, in conjunction with the rule eiej = −ejei,
i �= j, immediately yields
ei1 ∧ ei2 ∧ ... ∧ eir = ei1 ei2 ...eir . (2.32)
Besides, by means of the associativity of the outer product it is
(ei1 ei2 ...eir) ∧ (ej1 ej2 ...ejs) =
(ei1 ∧ ei2 ∧ ... ∧ eir) ∧ (ej1 ∧ ej2 ∧ ... ∧ ejs) =
ei1 ∧ ei2 ∧ ... ∧ eir ∧ ej1 ∧ ej2 ∧ ... ∧ ejs =
ei1 ei2 ...eirej1 ej2 ...ejs ,