Conformal Geometric Algebra in Stochastic Optimization Problems ...

48 CHAPTER 2. GEOMETRIC ALGEBRA
due to the associativity of the geometric product. The set {z ′ 1...k } is a new frame
for A 〈k〉 built of the reflected versions (see page 27) of the vectors {z1...k }. The
orthogonality is retained as can be seen by
z ′ i · z ′ j = (−nzin) · (−nzjn)
= 1
2
�
nzinnzjn + nzjnnzin
= n(zi · zj)n = δij.
In conclusion, the operation nA 〈k〉 n, where n is contained in A 〈k〉 , amounts to
the reflection of the entire basis, thereby letting the subspace invariant. Also in
general, if n is not or partly contained in A 〈k〉 , the operation nA 〈k〉 n represents
the reflection of A 〈k〉 in n. The specific property of blades that particular operations
as the reflection affect the basis is called outermorphism and is the subject of section
2.3.4.
Technically speaking, a vector is in fact a 1-blade. It is interesting that in this
regard a certain property of vectors can be attributed to blades as well - blades
square to scalars. The proof employs the previous proposition.
Corollary 2.10
For every blade A 〈k〉 it holds that
A 2 〈k〉
∈ �.
Proof: According to proposition 2.6, it exist k orthogonal vectors {z 1...k } so that
A 〈k〉 = z1z2 ...zk. Hence it may be written
A 〈k〉 A 〈k〉 = z1z2 ...zk z1z2 ...zk
k (k−1)
= (−1) 2 z 2 1 z 2 2 . .. z 2 k ∈ �.
This shows that all possible grades 0,2,4,...,2k of the geometric product A 2 〈k〉
vanish except the scalar part. This gives rise to the next corollary.
Corollary 2.11
The geometric product of a blade A 〈k〉 with itself coincides with the inner product,
i.e.
A 2 〈k〉 = 〈A 〈k〉 A 〈k〉 〉 = A 〈k〉 · A 〈k〉 .
The example
(a ∧ b) 2 = (a · b) 2 − a 2 b 2
demonstrates that it is possible that a blade squares to zero as well:
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