Editing Geometric algebra (section)

=== Reflection ===
[[Householder transformation|Simple reflections]] in a hyperplane are readily expressed in the algebra through conjugation with a single vector. These serve to generate the group of general [[rotoreflection]]s and [[rotation (mathematics)|rotations]].

[[File:GA reflection along vector.svg|200px|left|thumb|Reflection of vector <math>c</math> along a vector {{tmath|1= m }}. Only the component of <math>c</math> parallel to <math>m</math> is negated.]]

The reflection <math>c'</math> of a vector <math>c</math> along a vector {{tmath|1= m }}, or equivalently in the hyperplane orthogonal to {{tmath|1= m }}, is the same as negating the component of a vector parallel to {{tmath|1= m }}. The result of the reflection will be 
: <math> c' = {-c_{\| m} + c_{\perp m}} = {-(c \cdot m)m^{-1} + (c \wedge m)m^{-1}}
= {(-m \cdot c - m \wedge c)m^{-1}}
= -mcm^{-1} </math>

This is not the most general operation that may be regarded as a reflection when the dimension {{tmath|1= n \ge 4 }}. A general reflection may be expressed as the composite of any odd number of single-axis reflections. Thus, a general reflection <math>a'</math> of a vector <math>a</math> may be written
: <math> a \mapsto a' = -MaM^{-1} ,</math>
where
: <math> M = pq \cdots r</math> and <math> M^{-1} = (pq \cdots r)^{-1} = r^{-1} \cdots q^{-1}p^{-1} .</math>

If we define the reflection along a non-null vector <math>m</math> of the product of vectors as the reflection of every vector in the product along the same vector, we get for any product of an odd number of vectors that, by way of example,
: <math> (abc)' = a'b'c' = (-mam^{-1})(-mbm^{-1})(-mcm^{-1}) = -ma(m^{-1}m)b(m^{-1}m)cm^{-1} = -mabcm^{-1} \,</math>
and for the product of an even number of vectors that
: <math> (abcd)' = a'b'c'd' = (-mam^{-1})(-mbm^{-1})(-mcm^{-1})(-mdm^{-1}) = mabcdm^{-1} .</math>

Using the concept of every multivector ultimately being expressed in terms of vectors, the reflection of a general multivector <math>A</math> using any reflection versor <math>M</math> may be written
: <math> A \mapsto M\alpha(A)M^{-1} ,</math>
where <math>\alpha</math> is the [[automorphism]] of [[reflection through the origin]] of the vector space ({{tmath|1= v \mapsto -v }}) extended through linearity to the whole algebra.