Conformal Geometric Algebra in Stochastic Optimization Problems ...

2.2. BASIC CONCEPTS OF GA 49
Definition 2.6 ( Null blades ):
A blade A 〈k〉 is called a null blade if it squares to zero, or rather
A 〈k〉 · A 〈k〉 = 0.
Null blades can only occur in algebras of mixed signature. Since every blade has a
representation in terms of mutually orthogonal vectors, at least one of these vectors
must at the same time be a null vector in order to have a null blade.
Example 2.6 ( Null blades ):
Consider the mutually orthogonal vectors z1 = e1, z2 = e2 and z3 = e3 + e4 from
� 3,1 . Their outer product is A 〈k〉 = e1e2e3 + e1e2e4. The square of A 〈k〉 is
due to z2 3 = 0.
A 2 〈k〉 = −(z1z2z3)(z3z2z1) = −z 2 1z 2 2z 2 3 = −(1)(1)(0) = 0
Next to blades it exists another important class of multivectors in �p,q.
Definition 2.7 ( Versor):
A multivector is called a versor iff it can be expressed as the geometric product of
(invertible) non-null vectors.
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Note that with this definition any non-null blade is a versor at the same time.
Furthermore, bear in mind that
V ∈ �p,q versor �=⇒ V 2 ∈ �.
Another neat application of proposition 2.6 effortlessly shows that
A 〈k〉 bA 〈k〉 = z1z2 ...zk b z1z2 ...zk
=
k (k−1)
(−1) 2 z1z2 ...zk b zkzk−1 ...z1
=
k (k−1)
(−1) 2 z1(z2(...(zk−1(zk b zk)zk−1)...)z2)z1
is (apart from a scalar factor since z 2 i �= 1, i ∈ [1,k] � ) nothing but a series of
reflections of b in the vectors {z 1...k }. The result of A 〈k〉 bA 〈k〉 is therefore a vector
as well, i.e.
A 〈k〉 bA 〈k〉 ∈ � p,q . (2.40)
It is known from linear algebra that a vector a can be decomposed with respect
to its component a � inside and the component a⊥ outside a certain subspace B 〈l〉
, i.e. a = a � + a⊥. A decomposition with respect to a vector - a 1-blade - is
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