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Introduction

    Geometric algebra provides a practical alternative to conventional 3D vector methods which extends far more readily to higher dimensions. It also provides a coordinate independant symbolic geometry (an "algebra of directions") extendable into a geometric calculus of profound relevance to areas as diverse as quantum physics and computer vision. Although a notable advantage of geometric algebtra is coordinate independance, we will   initially take a coordinate (basis) based approach here since this is likely to be the most practicable in a programming context.
    The purpose of this section is to provide a concise but comprehensive introduction and broad reference for geometric algebra for those interested in it as a powerful computational and theoretical resource that spans and unifies a diverse range of fields. Fuller more formal treatments exist elsewhere and this document can serve as a primer for tackling such works. It assumes familiarity with "conventional" 3D constructs such as vectors and matrices and such basic mathematical functions as cosq and e^x.  Mathmatical notations used are defined in the glossary.
    This treatment favours the contractive "computer scientist's" inner product a¿b over the semisymmetric "physicist's" inner product a.b where possible because this is arguably the more fundamental and has certain functional advantages.

    Any reader who enjoys mathematics but has not encountered multivectors before should prepare themselves for a treat.

    When first encountering multivectors they can seem bewildering, with a plethora of products and an overload of operators provided for their manipulation. But familiarity breeds respect. Given that multivectors are the "language" of (dynamic) geometry and by implication of nature itself, a half-dozen new symbols does not seem excessive.

    In this chapter we will dscribe the mathematical manipulations of multivectors, their various products and conjugations and operations such as exponentiation and logarithms, projections, and intersections. Of necessity some of this will be mathematically intensive and the reader is is encouraged to "skim" through such material rather than absorbing every product, conjugation, normalisation technique and exponential computation strategum on their first pass. The "point" of multivectors is what you can do with them, and this is addressed in the later sections
    In the Multivectors Programming chapter we describe how to impliment multivectors of both low and high dimension N in C/C++.
    In the Multivectors as Geometric Objects we will see some of the applications of multivectors in the elegant representation and manipulation of N-dimensional spheres, planes, lines and conics.
    In the Multivectors as Transformations we will see how multivectors can also be used to transform, distort, and displace such geometric constructs.
    In Multivector Arcana we cover some more esoteric mathematical aspects of multivectors of less general interest.
    In later sections we cover Multivector Calculus and the uses of multivectors in physics, in particular Relativity and Quantum Mechanics.

Multivectors
"And therefore in geometry (which is the only science that it hath pleased God hitherto to bestow on mankind), men begin at settling the significations of their words; which settling of significations, they call definitions, and place them in the beginning of their reckoning." ---  Thomas Hobbes, Leviathan

    Geometric algebra is essentially a set of arithmetical techniques for manipulating N-dimensional vectors and to see how to properly multiply (and divide!) vectors we must first generalise the concept of vectors and scalars.
    Given k linearly independant N-dimensional vectors a₁,a₂,..,a_k from a vector space U^N [  we are interested initially in U^N=Â^N, the space of real coordinate N-D vectors, but will refer to U^N to emphasise applicability to alternate (eg. nonEuclidean) spaces which are of interest to us, particularly with regard to relativistic physics. By "linearly independant" we here mean that no one of the a₁,a₂,..a_k can be expressed as a real-wighted sum of the others ] their outer product a_k = a₁Ùa₂...Ùa_k (not to be confused with the 3D vector "cross" product)  is known as a blade of grade (aka. step or degree) k, or a k-blade. The fundamental rules for Ù are
     antisymmetry: aÙb = -bÙa ( and so aÙa=0) for any vectors a and b ;
    linearity: aÙ(b+c) = aÙb + aÙc ; and
    associativity: aÙ(bÙc) = (aÙb)Ùc.
     A k-blade a_k can be thought of as representing an orientated and scaled k-dimensional subspace of U^N, one in which all the vectors satisfy aÙa_k = 0. We say that a₁,a₂,..,a_k are a k-frame for this subspace.
    A linear "weighted additive" combination of k-blades is known as a k-vector. An example 4D 3-vector is ½e₁Ùe₃Ùe₄ + (Ö7)e₁Ùe₂Ùe₃ .
    A 0-vector is a 0-blade is a scalar. We refer to a k-blade as proper if k>=1.
    A 1-vector is a 1-blade is a conventional vector.
    2-blades can be considered geometrically as directed areas (ie. a plane and a signed scalar).
    2-vectors (aka. bivectors) are sums of scaled 2-blades and need not "reduce" to a single 2-blade for N>3.
     For N£3, any k-vector is a k-blade. This is true only for k<2 when N > 3 .
[ Proof :  ae₁Ùe₃+be₃Ùe₁+ge₁Ùe₂ = (e₁+(a/g)e₃)Ù(ge₂-be₃) if g¹0, (ae₂-be₁)Ùe₃ else . All 3D 3-vectors are multiples of e₁Ùe₂Ùe₃    .]

    Consider the set Â_N (aka. G_N and Cl_N in the literature) of all linear combinations (with real "coefficients" or "wieghtings") of k-vectors for 0£k£N, our 1-vectors being taken from the N-dimensional vector space Â^N. Clearly Â^N Ì Â_N (or, more properly, is represented within Â_N) and infact Â_N has dimension å_k=0^N NC_k = 2^k, there being ^NC_k º N! (N-k)!^-1 k!^-1 distinct k-blades in N-dimensions.

    We refer to the elements of Â_N as multivectors An example Â₃ multivector is 3+e₁-4e₂+½e₁Ùe₂+(Ö7)e₂Ùe₃ + pe₁Ùe₂Ùe₃. We can thus think of a general N-D multivector as an arbitary real-weighted combination of 2^N distinct basis blades. [  Mathematically, one can take the "blade coefficients" from any field or "number space". It is the utilisation of  - essentially identifying the blade "coefficients" or "coordinates" with "scalars" (0-vectors) - that distinguishes "geometric algebra" from more general Clifford algebras of only passing concern here. If we allow "complex number" blade coeeficients, for example,  we obtain a space C_N of dimension 2^N+1 . ]
    Multivectors can thus be represented with 2^N dimensional 1-vectors, with respect to a given set of N linearly independant N-D basis 1-vectors e₁,e₂,...,e_N and this tells us how to add and subtract them, but not how to multiply and divide them. For that we will need the "geometric product" and its associated "subproducts".

Conflicting Terminologies
    Some authors such as Pavsik use the term "k-vector" for what we will call a k-blade and "multivector" for our k-vector (ie. a single-graded multivector), adopting the term polyvector for our multivector.

Notations and Coordinates
    A general N-D multivector is thus the sum of a scalar, a 1-vector, a 2-vector,..., and an N-vector but we will frequently be interested in pure k-blades or k-vectors and will benefit from a notation that that distinguishes such from general multivectors.  We will use the following fonts to denote geometric algebra elements

Font	Represents	Also known as
a	Proper blade
a	General multivector
a	0-vector	Scalar
a	1-vector	Vector
a	2-vector	Bivector
a	(N-2)-vector	Pseudovector
a	(N-1)-vector	Pseudoscalar

Blades
    Given an orthonormal basis (ie. a Cartesian coordinate frame) consisting of N orthonormal N-D 1-vectors {e₁,e₂,...,e_N} for Â^N, we can consider multivectors as elements of Â^2N, ie. as 2^N-dimensional 1-vectors,
    We use the notation e_ij..m º e_iÙe_j....Ùe_m     where i,j,..m are distinct indices. [  If we relax the distinctness restriction in the case of N indices we have e_ij..m = e_12..Ne_ij..m where scalar e_ij..m is the traditional "alternating tensor" returning ±1 or 0 according as whether ordered set {i,j,..,m } is an even/odd, or not a, permutation of ordered set { 1,2, .. N } ]
    If all 2^N coefficients of the basis blades arising from a given orthonormal basis are integers ( eg. 5 + 3e₁ - 7e₂Ùe₃) then we will say the multivector is integer-coefficiented with regard to that basis. If the coeffients are further restricted to { ±1 , 0 } then the resulting 3^2N "discrete" multivectors are said to be Mantheyian with regard to the basis.

Specific subcomponents
    We will write a^ij..m or a_[ij..m] for the coefficient of e_ij..m in multivector a. . [  We will seldom have to raise individual components to integer powers and this raised suffix notation has definite advantages ]
    We will refer to the integer ½(2^j+2^k+...+2^m) as the bitwise enumerator for the extended basis blade e_jk..m. The ½ arises because we labelled our first frame vector e₁ rather than e₀. We will denote coordinates or blades indexed by a particular bitwise enumerator n by _[.n.] . Thus eg. e_[.0.]=1 ; e_[.1.]=e₁ ; e_[.6.]=e₂₃ and so on and we have a = å_k=0^2N-1 a_[.k.] b_[.k.] .
    We will use the notation a_k to indicate a k-vector, a_k a k-blade, and the notation a_<k> to indicate the k-vector component of a general multivector a (with a_<k>=0_<k>=0 for k>N) . The notation a_<i,j,..m>º a_<i>+ a_<j>+ .. + a_<m> will prove useful.
    We say a is pure or k-pure if a_<k> = a. By an impure k-vector we mean a multivector having a nonzero k-vector component and possible nonzero j-vector components for j<k.
    If we wish to specify a multivector as having only components of grades 2,3 and 5, for example, we will refer to a <2,3,5>-vector. A <k>-vector is thus a pure k-vector. A <0,1>-vector is sometimes known as a paravector, particularly when N=3.
    We will use the notations a_<+> º a_{<0,2,..,2*[N/2|>} to denote the multivector obtained by taking only the even grade components of a, and a_<-> º a_{<1,3,..,2*[N/2|+1>} for the odd grade components.  

Zeroes
    The degenerate (zero magnitude) k-blade is denoted by 0_k.
    All degenerate blades are considered equivalent ( 0 = 0₁ = 0₂ = ....0_N = 0 ) , and this zero is the sole equivalence. Note that within a computational context (finite precision arithmetic) we will typically be checking for proximity rather than equivalence with zero. It is often worth keeping track of the "grades" of expected zeros.

    We are freuwently interested in issues such as whether the square of a multivector is a pure scalar but in practice this may mean checking that any residual nonzero coordinates are a "negligable" proportion of the scalar part, which can be problematic if the scalar part is also near zero.
    We define the sparsity of a multivector with respect to a given basis as the number of zero coefficients (coordinates) in its representation in that basis.

Inverse frames
    Suppose we have a possibly non-orthonormal linearly independant basis of N N-D 1-vectors providing a coordinate frame (ie. a set of axies) E=(e₁,e₂,...,e_N) for U^N. We can construct a reciprocal or inverse frame (e¹,e²,...,e^N) so that eⁱ.e_j = 1 when i=j and 0 else where . is the traditional scalar ("dot") product of two 1-vectors.
    If E is orthogonal then provided no e_i²=0 we can set e^k º e_k^-2e_k . More generally we require e^k º (-1)^k-1(e₁Ù..e_k-1Ùe_k+1Ù...e_N) i^-1 where i=e_12..N .   However, this may not make much sense to the reader till he is more familiar with Ù and the pseudoscalar i discussed later.
    We define a notation e^ij..m º eⁱÙe^jÙ...e^m .

    If E is orthogonal (ie. e_i.e_j=0 for i¹j) then (using the geometric product defined below) we have e^ke_k = 1 (provided e_k²¹0) ; but in general e^ke_k has a non-zero 2-blade component because e^kÙe_k ¹ 0 .
    E induces both the coordinate expression x = å_i=1^N xⁱe_i     [ with xⁱ º eⁱ.x ], and the reciprocal coordinate expression x = å_i=1^N x_ieⁱ     [ with x_i º x.e_i ]. eⁱ is the 1-vector geometric multiplier that "seperates"  1-vector x into xⁱ + å_{j ¹ i} x^jeⁱÙe_j .

    In Â^N : (i) an orthonormal frame is self-inverse ( e_i = eⁱ ; x_i = xⁱ ) ; (ii) a general frame, expressed as an N×N matrix E with respect to a fixed orthonormal frame F=(f₁,f₂,...f_N) in conventional manner via ( e_i = å_i=1^N E_jif_j ) has as its reciprocal frame the frame having matrix (E^-1)^T = (E^T)^-1   with respect to F, ie. the inverse transpose.

    Letting E_ij º e_i¿e_j  and E^ij º eⁱ¿e^j ,  we have x_j = å_i=1^N E_ijxⁱ     ;      x^j = å_i=1^N E^ijx_i .
    The N×N symmetric matrices {E_ij} and {E^ij} are related by {E^ij} = {E_ij}^-1   where ^-1 is  the conventional matrix inverse.

Inverse Frame Units
    If frame vectors are assigned units, e₁ having length 5 m say, then e¹ must be regarded as having "length" 5^-1 _meteri so that e¹¿e₁ = 1 m⁰ is dimensionless. Coordinates _xui =eⁱ¿x are then unitless while reciprocal coordinates x_i = e_i¿x have units m². x_i 2 the inverse

Extended inverse frames
    Given an extended basis {e_[.i.] : 0£i<2^N } we can construct an extended pureblade inverse frame {e^[.i.]} which satisfies e^[.i.]_* e_[.j.] º (e^[.i.]e_[.j.])_<0> =  1 Û i=j , 0 else .

The Geometric Product
"He who can propery define and divide is to be considered a god." --- Plato

    To make Â_N a linear space we require a "multiplication" with the following properties:
    a(bc) = (ab)c     (associativity)
    a(b+c) = ab + ac ; (b+c)a = ba + ca     (distributivity)
    aa = Sig(a) where Sig(a) is a scalar for all 1-vector a     (contraction).
    Of principle interest here is the contraction Sig(a) = e|a|². where e (the signature of a) is either ±1 or 0 and |a| is the conventional magnitude ("length") of 1-vector a. A vector is null if a²=0.   
    We write Â^p,q,r for a vector space having orthogonal basis {e₁,e₂,...e_N} where N=p+q+r and Sig(e_i) = 1 for 1£i£p ; -1 for p<i£p+q ; 0 for p+q<i£N .
    We write Â_p,q,r for the associated geometric algebra. We write Â_p,q (aka. Cl_p,q) as an abbreviation for Â_p,q,0 and Â_N as an abbreviation for Â_N,0,0 .

    We define the geometric product of any a by a scalar b in the obvious "coordinatewise" commutative manner (ba)_[ij..m] =    (ab)_[ij..m] º b(a_[ij...m]) .
     We define the geometric product of two 1-vectors by ab º a.b + aÙb where a.b is the conventional Â^N (or U^N) vector "dot" product. We can then extend this definiton by means of the associativity and contraction rules to higher grade blades and hence (by distributivity) to multivectors generally. The geometric product is noncommutative, but this is actually an assett; the "degree" of noncommutativity of the geometric product of two multivectors being a measure of their orthogonality. A unit multivector is a multivector satisfying (aa)_<0>=±1.

Â₂
    We can tabulate the geometric product for Â₂ with respect to a basis for Â₂ derived from an orthonormal basis {e₁,e₂} for ².

ab for Â2		a
		1	e1	e2	e12
	1	1	e1	e2	e12
b	e1	e1	1	e12	-e2
	e2	e2	-e12	1	e1
	e12	e12	e2	-e1	-1

[ Proof :  e₁₂e₁ = (e₁Ùe₂)e₁ = -(e₂Ùe₁)e₁ = -(e₂e₁)e₁ = -e₂(e₁e₁) = -e₂ . Other products likewise.  .]
    We note in passing that the subspace Â_{2 +} consisting of all 2D multivectors having no 1-vector component, ie. the space of multivectors of the form a + be₁₂   is closed under the geometric product and is isomorphic to the complex number space C,  as is Â_0,1.

Â_1.1
    If e₁²=1 and e₂²=-1 then (e₁Ùe₂)² = 1 and the even subalgebra Â_{1,1 +} generated by 1 and (e₁Ùe₂) is isomorphic to the hyperbolic numbers x+yh where h²=1.
      These are less well studied than complex numbers and rarely implimented in code libraries. However it is easy to generalise a complex number class to allow for a plusquare or null i. The hyperbolic number x+yh has inverse (x-yh)(x²-y²)^-1 provided x¹±y . Though we can express x+yh in "hyperbolic polar form" r(qh)^↑ where r=|x²-y²|^½ and q =  tanh^-1(xy^-1) if |x|<|y| ; or  tanh^-1(yx^-1) if |y|<|x| , the inverse hyperbolic tangent providing q must be carefully defined since hyperbolic numbers divide naturally into four "regions" or "quadrants" seperated by the lines y=x and y=-x according as to which of |x| and |y| is larger and the sign of the larger coordinate. We will refer to hyperbolic numbers with x>0 and x>|y| as being in the (1)-quadrant since this is the quadrant containing 1. Hyperbolic numbers in this quadrant remain in it when raised to any power or exponentiated.
    We can express ay (1)-quadrant hypercomplex as r(qh)^↑ for real r³0 and q Î (-¥,¥) . Whereas the complex polar angle parameter can be restricted to (-p,p] and maps all 2-D directions, the hyperbolic polar angle parameter is unbounded and yet only maps a quarter of the 2-D directions. We cannot recover x and y from r and q, as we can for complex numbers, without also knowing which quadrant we are in. Multiplication with h flips x and y and so corresonds to reflection in x=y , mapping the (1)-quadrant into the (h)-quadrant, the (h)-quadrant into the (-1)-quadrant, and so on.   Computationally, we must specify the quadrant "mapped" by q as well as r and q, which requires an additional two bits.

Â₃
    We can tabulate the geometric product for Â₃ with respect to a basis derived from a standard orthonormal basis {e₁,e₂,e₃} for ³.

ab for Â3		a
		1	e1	e2	e3	e23	e31	e12	e123
	1	1	e1	e2	e3	e23	e31	e12	e123
	e1	e1	1	-e12	e31	e123	e3	-e2	e23
	e2	e2	e12	1	-e23	-e3	e123	e1	e31
b	e3	e3	-e31	e23	1	e2	-e1	e123	e12
	e23	e23	e123	e3	-e2	-1	e12	-e31	-e1
	e31	e31	-e3	e123	e1	-e12	-1	e23	-e2
	e12	e12	e2	-e1	e123	e31	-e23	-1	-e3
	e123	e123	e23	e31	e12	-e1	-e2	-e3	-1

and similar geometric multiplication tables can be constructed for higher N.
    What is happening here is e_[.j.]e_[.k.] = ± e_{[.j  ^  k.]} where  ^  denotes bitwise XOR ("exclusive or") and the sign is determined both by how many commutations we have to do to bring the common e_i together, and the signatures of the common e_i.

    Writing i º e₁₂₃ º e₁Ùe₂Ùe₃ = e₁e₂e₃ we see from the above table that i commutes with all multivectors and satisfies i² = -1.
    We also observe that aÙb = i(a×b) = (a×b)i where a×b is the conventional ³ vector "cross" product and that ia spans the plane normal to a. We also have aÙbÙc = (a.(b×c))i.
    We note in passing that the subspace Â_{3 +} consisting of all 3D multivectors having no odd grade component, ie. the space of multivectors of the form a + be₂₃ + ge₃₁ + de₁₂ is closed under the geometric product and is isomorphic to the quaternion space Q, as is Â_0,2 . a + bi + gj + dk.

Biquaternions
    We further note that a general Â₃ multivector can be uniquely expressed as (a + a₂) + (b + b₂)e₁₂₃ where a,b are scalars and a₂,b₂ are pure Â₃ bivectors. An alternative biquaternion (aka. complex four-vector aka. Pauli spinor) representation of Â₃ sets i=i=e₁₂₃ , s₁=e₁ , s₂=e₂ , s₃=e₃ (satisfying s_i s_j=e_ijki s_k). The biquaternion (a₀+b₀i) +(a₁+b₁i)e₁ +(a₂+b₂i)e₂ +(a₃+b₃i)e₃ is equivalent to the Â₃ multivector a₀ + a₁e₁+a₂e₂+a₃e₃ + b₁e₂₃+b₂e₃₁+b₃e₁₂ + b₀e₁₂₃ . The product of two 3D 1-vectors is usually taken in this context to be ab º a.b + i(a×b) which is equivalent to the Â₃ geometric product since i(a×b)=aÙb.

Pseudoscalars
    The N-vectors from a given U^N are all equivalent apart from magnitude ("scale","volume") and sign ("handedness"). They are accordingly known as pseudoscalars. Conversely a nonzero pseudoscalar "spans" (and can be thought of as representing) U^N .
    We will use the font a to denote a multivector viewed as a pseudoscalar.   
    Let i º i_N be the unit pseudoscalar e_12..N for Â_N. i satisifes i ² = (-1)^½(N-1)N and commutes with all multivectors if N is odd. For even N we have ia_k = (-1)^ka_ki so that i (anti)commutes with (odd)even blades.
    We say a multivector is central if it commutes with all other multivectors. For even N, only scalars are central but for odd N any <0;N>-multivector (scalar plus pseudoscalar) is central.
    An (N-1)-vector is sometimes refreerd to as a pseudovector but we favour the term hyperblade here.
    Taking the geometric product of a multivector with i maps k-blades to (N-k)-blades and vice versa. In particular it maps scalars to pseudoscalars (and vice versa) and vectors to pseudovectors (and vice versa).
    Note that a k-blade acts as a pseudoscalar when acting upon multivectors wholly contained within the space it spans. The geometric product of a pseudoscalar i with a blade a_k contained in the subspace spanned by i spans the subspace of i complimentary (orthogonal) to subspace a_k. In particular, the geometric product of any blade with itself is a scalar.
    The signature   of a blade b_k, e_bk , is the sign of b_k² (or zero if b_k² = 0 in which case the blade is said to be null).

Duality

    We define the dual of a multivector a with respect to a pseudoscalar i spanning a space containing a by
    a^* º ai^-1 = a¿i^-1
[  Where ¿ is the contractive inner product defined below. Some authors favour ai, but if i is a unit psuedoscalar the difference is only one of sign. ]
    a^* spans the subspace of i "perpendicular" to pureblade a.
    If b is an unmixed (ie. odd or even) multivector, it will either commute or anticommute with i. For odd N, the pseudoscalar commutes with everything and we have (a^*)b = a(b^*) = (ab)^* .
    For even N, i (anti)commutes with (odd) even multivectors and we have (a^*)b = a(b^#*) = (a(b^#))^* where ^# is the grade involution conjugation defined below.

    In the presence of a standard basis for Â_N, computing a^* for the unit pseudoscalar i=e_123..N is a computationally trivial "shuffling" of coordinates requiring no numeric computations. For the bitwise ordering we have (ai^-1)_[.i.] = ± a_{[.(i XOR (2^N-1)).]} where the actual sign depends upon N and the bitwise parity of i.

    The inverse dual or undual is defined by a^-* º a^*i² = ai so that (a^-*)^* = a.

Matrix representations


Â_p.q.r in Â_2^N×2^N
    With regard to a particular extended basis, an N-D multivector a can be expressed as a 2^N dimensional real 1-vector. But as a function mapping multivectors to multivectors a(x) = ax , ie.. a transform of 2^N-D 1-vectors, a can also be represented as a 2^N×2^N matrix.
    Taking a=a⁰+a¹e₁+a²e₂+a¹²e₁₂ in Â_p,q,r with p+q+r=2, for example, we have
    (a⁰+a¹e₁+a²e₂+a¹²e₁₂)(x⁰+x¹e₁+x²e₂+x¹²e₁₂)
        = (a⁰x⁰+e₁a¹x¹+e₂a²x²-e₁e₂a¹²x¹²) + (a¹x⁰+a⁰x¹+e₂a¹²x²-e₂a²x¹²)e₁ + (a²x⁰+a⁰x²-e₁a¹²x¹+e₁a¹x¹²)e₂ + (a⁰x¹²+a¹x²-a²x¹+a¹²)e₁₂       
which we can express as

æ	a0	e1a1	e2a2	-e1e2a12	ö		æ	x0	ö
ç	a1	a0	e2a12	-a2	÷		ç	x1	÷
ç	a2	-e1a12	a0	e1a1	÷		ç	x2	÷
è	a12	-a2	a1	a0	ø		è	x12	ø

    Note that the leftmost column of the matrix representation of a contains the 1-vector representation of a, and that the matrix is symmetric apart from occasional sign changes determined by both the position in the matrix and the signatures of the basis 1-vectors.
    Note also that the matrix trace (sum of leading diagonal elements) of the matrix representatrion of a is the frame-invariant scalar 2^Na⁰ = 2^Na_<0> .
    The geometric product ab corresponds to conventional matrix multiplication of the matrix representations of a and b. Multivector 1 has as its matrix representation the 2^N×2^N identity matrix.
    If the pseudoscalar is central (all commuting) then we can express each multivector as a 2^N-1 complex 1-vector and obtain a 2^N-1×2^N-1 complex matrix, which is half the size as a 2^N×2^N real matrix.
    Matrix representations of multivectors provide an alternate model for those who have philosophical difficulties with the concept of adding "different grade" blades to form a composite "entity". Geometric (multivector) algebra becomes the algebra of a particular "form" of matrix, requiring only standard matrix multiply and inversion techniques. Such an approach is computationally profligate, but can sometimes provide alternative insights and quick-and-dirty programming applications exploiting existing matrix suites.

    Far more compact matrix representors for multivectors are typically available. The following are all "maximally efficient" in that they require precisely 2^N real scalar parameters to hold a general N-D multivector.

Â₂ in Â_2×2

1	=	æ	1	0	ö	e1	=	æ	1	0	ö	e2	=	æ	0	1	ö	e12	=	æ	0	1	ö
		è	0	1	ø			è	0	-1	ø			è	1	0	ø			è	-1	0	ø

satisfy the required e₁²=e₂²=1 ; e₁₂=e₁e₂=-e₂e₁ .

Â_1.1 in Â_2×2

1	=	æ	1	0	ö	e1	=	æ	0	1	ö	e2	=	æ	0	-1	ö	e12	=	æ	1	0	ö
		è	0	1	ø			è	1	0	ø			è	1	0	ø			è	0	-1	ø

satisfy the required e₁²=1 ; e₂²=-1 ; e₁₂=e₁e₂=-e₂e₁ .

Â₃ in C_2×2

    Â₃ has a "biquaternian" representation with Hermitian 2×2 complex matrices

1=1

=

æ

1

0

ö

; e1= s1

=

æ

0

1

ö

; e2= s2

=

æ

0

-i

ö

; e3= s3

=

æ

1

0

ö

è

0

1

ø

è

1

0

ø

è

i

0

ø

è

0

-1

ø

which provide the Pauli algebra if we set i=e₁₂₃=i (so that s₁ s₂ s₃=i s₃² = i 1 )

ab		a
		1	s1	s2	s3
b	1	1	s1	s2	s3
	s1	s1	1	-i s3	+i s2
	s2	s2	+i s3	1	-i s1
	s3	s3	-i s2	+i s1	1

Note that s_i^D=- s_i while 1^D=1 , traditional 2×2 matrix adjoint ^D corresponding to a biquaternian "vector conjugation".

    The s_i and 1 act as a basis for the full   C_2×2 algenra of complex 2×2 matrices since

æ	a	b	ö	= ½(a+d)1 + ½(b+c) s1   + ½i(b-c) s2 + ½(a-d) s3
è	c	d	ø

    Each element of the special unitary group SU(2) of all C_2×2 unitary (AA^†=1) matrices having unit positive determinant can be expressed via U=(½iå_j=1³ q_je_j)^↑ for three real scalar parameters q_j. e₁,e₂, and e₃ are then refered to as the generators of SU(2) so we can view SU(2) geometrically as the space of exponentiated Â₃ bivectors.
    SU(2) is isomorphic to the group SO(3) of all orthogonal (AA_transp=1) Â_3×3 matrices, with U (U)

æ z	x-iy	ö	U-1
è x+iy	z	ø

equivalent to Ax where x=(x,y,z)_transp, yielding the quaternian representation Â_{3 +} @ SO(3).

Â_3.1 in Â_4×4

1=1	=	æ	1	0	0	0	ö	; e1	=	æ	1	0	0	0	ö	; e2	=	æ	0	1	0	0	ö	; e3	=	æ	0	0	1	0	ö	; e4	=	æ	0	0	-1	0	ö
		ç	0	1	0	0	÷			ç	0	-1	0	0	÷			ç	1	0	0	0	÷			ç	0	0	0	-1	÷			ç	0	0	0	1	÷
		ç	0	0	1	0	÷			ç	0	0	-1	0	÷			ç	0	0	0	1	÷			ç	1	0	0	0	÷			ç	1	0	0	0	÷
		è	0	0	0	1	ø			è	0	0	0	1	ø			è	0	0	1	0	ø			è	0	-1	0	0	ø			è	0	-1	0	0	ø

satisfy the required e₁²=e₂²= e₃²=-e₄²=1 ; e_ie_j=-e_je_i for i¹j . [ Due to Lounesto.] Note that the 1-vector representors are traceless and have determinant ±1. Transpose negates only e₄ and coresponds to geometric Hermitian conjugation.

Â_4.1 in C_4×4

1=	æ	1	0	0	0	ö    ;	e1=	æ	0	0	0	i	ö    ;	e2=	æ	0	0	0	1	ö ;	e3=	æ	0	0	i	0	ö ;	e4=	æ	0	0	1	0	ö ;	e5=	æ	-i	0	0	0	ö
	ç	0	1	0	0	÷		ç	0	0	i	0	÷		ç	0	0	-1	0	÷		ç	0	0	0	-i	÷		ç	0	0	0	1	÷		ç	0	-i	0	0	÷
	ç	0	0	1	0	÷		ç	0	-i	0	0	÷		ç	0	-1	0	0	÷		ç	-i	0	0	0	÷		ç	1	0	0	0	÷		ç	0	0	i	0	÷
	è	0	0	0	1	ø		è	-i	0	0	0	ø		è	1	0	0	0	ø		è	0	i	0	0	ø		è	0	1	0	0	ø		è	0	0	0	i	ø

satisfy the required e₁²=e₂²=e₃²=e₄²= -e₅²=1 ; e_ie_j=-e_je_i  for  i¹j . Once again, the 1-vector representors are traceless and have determinant ±1. Note that e₁₂₃₄₅ = e₁e₂e₃e₄e₅ = i1 = i with e₁₂₃₄=-ie₅ .
    Setting e₅'=ie₅ provides a C_4×4 representation of Â₅ while setting e_i'=ie_j for j£4  provides one for Â_0,5.
    Setting g⁰=ie₅ , g¹=-ie₁ , g²=-ie₂ , g³=-ie₃ provides a "Dirac matrix" basis for Â_1,3 , inefficient in that it requires 16 complex values to represent a 4D multivector having 16 real coordinates.
    Note that matrix transpose coresponds to negation of e₁ and e₃ whereas matrix complex conjugation negates e₁,e₃, and e₅ so that matrix Hermitian conjugation A^{T^} negates only negative signatured e₅ and so coresponds to geometric Hermitian conjugation defined below. Negation of the on-lead-diagonal blocks is provided by e₅₌ and negates 1 and e₅.

    We can construct matrices having 1 in the first to fourth entries of the first column (and zeroes elsewhere) respectively as

• u   =   -(i-e₅)(i+e₁₂)   =   (-1+ie₅)(-1+ie₁₂)   =   (1-ie₅)(1-ie₁₂)   =   (1+e₁₂₃₄)(1+e₃₄₅) [ an absorber or emmitter of e₃₄₅, e₁₂₃₄ and e₁₂₅] ;
• e₁₃u = -e₁₄₅u     ;
• ie₃u   =   e₁₂₄₅u   =   -e₃₅u     ; and
• ie₁u   =   e₂u=e₄u=-e₂₃₄₅u =e₁₂₄₅ (-e₁₄₅u)   =   _bl2v(e₁₅)u

    We define the contractive inner product (aka. Lounesto inner product) (¿ or û ) by
    a_k¿b_l º ( a_kb_l)_<l-k>       for l³k    ,    0  else   
where a_k,b_l are blades of any grade; and thence extending over multivectors by insisting on bilinearity.
    ¿ is neither associative nor commutative (symmetric). In particular, (a¿a)=a² so  (a¿a)¿b=a² b but a¿(a¿b) = 0 .
    For k-vectors with k>1, contraction with a 1-vector a corresponds to orthogonal projection into subspace a^* so
a¿b is the "component factor" of blade b perpendicular to 1-vector a   .

    We have the following properties:

• a¿b = ab    ;     b¿a = ab_<0>
• a¿b = ½(ab+ba)
• ab = a¿b + aÙb
• a¿b_k = ½(ab_k - (-1)^kb_ka) = ½(ab_k - b_k^#a)     k³0
• a¿(bÙc) = (a¿b)Ùc + b^*Ù(a¿c)
• a¿(b¿c) = (aÙb)¿c         [ the ¿ for Ù swapping rule ].     Þ a¿(a¿b)   = 0
• a¿(b¿j) = (aÙb)¿j     where j is any pseudoscalar spanning a subspace containing b.
• a¿(b₁Ù...Ùb_k) = å_i=1^k (-1)ⁱ⁺¹(a¿b_i) (b₁Ù..Ùb_i-1Ùb_i+1Ù...Ùb_k)     [ the expanded k-blade contraction rule]. Examples include:
      a¿(bÙc) = (a¿b)c - (a¿c)b       [ the expanded bivector contraction rule] giving a¿(bÙc) = - a×(b×c) in Â₃ .
      a¿(bÙcÙd) =(a¿b)(cÙd) - (a¿c)(bÙd) + (a¿d)(bÙc)
• (a₁Ù...Ùa_r)¿(b₁Ù...Ùb_s) = (a₁Ù...Ùa_r-1)¿(a_r¿(b₁Ù...Ùb_s)) = å_i=1^s (-1)ⁱ⁺¹(a_r¿b_i)(a₁Ù...Ùa_r-1) ¿(b₁Ù...Ùb_i-1Ùb_i+1Ù...b_s)
  [  note ¿ rather than . and that expression is vanishingly true for r>s ]
      Whence (a_kÙ...Ùa₂Ùa₁)¿(b₁Ùb₂Ù...Ùb_k) = Det([a_{i j}]) where a_{i j} = a_i¿b_j .
      In particular,     (a₁Ùa₂)¿(b₁Ùb₂) = a₁¿(a₂¿(b₁Ùb₂)) = (a₂¿b₁)(a₁¿b₂) - (a₁¿b₁)(a₂¿b₂)
• a¿(b₂¿c₃) = (aÙb₂)¿c₃     where b₂ is a 2-vector and c₃ is a 3-vector.
• (aÙb)(a+b) = (a²+a¿b)(a-b)     when a²=b²
• a¿(b_kÙc) = (a¿b_k)Ùc + (-1)^kb_kÙ(a¿c).     called the expanded contraction rule with special case a¿(bÙc) = (a¿b)c - bÙ(a¿c).

    ¿ is sometimes known as left-contraction or onto-contraction as opposed to the right- or by-contraction defined by
    a_k ë b_m º ( a_kb_m)_<k-m>       for k³m ; 0 else
or equivalently by aëb = (a^§¿b^§)^§ where ^§ is the reverse operator defined below.

The semi-commutative inner product
    Some authors favour the semi-symmetric or semi-commutative inner product (aka. Hestenes inner product) (.) defined by
    a.b_k º b_k.a º 0 where b_k is any blade  ;  a_k.b_m º ( a_kb_m)_<|k-m|> where a_k,b_m are proper (nonscalar) blades;
and thence extending over multivectors by inisiting on bilinearity. The result is neither associative nor commutative ("symmetric").  It is "semi-symmetric" in that a_j.b_k = (-1)^j(k-j)b_k.a_j for pure multivectors a_j,b_k with j£k.
    Note that scalars (0-blades) a,b satisfy aÙb = ab ; a¿b = a.b = 0 so we can think of scalars as "self-orthogonal".
    We obtain the identities

• a.b_k = (-1)^k+1b_k.a .
• b_ka = b_k.a + b_kÙa     for kÎ{1,2}
• a.b_k = ½(ab_k - (-1)^kb_ka)    ;     b_k.a = (-1)^k+1a.b_k .
• a.(b_kÙc) = (a.b_k)Ùc + (-1)^kb_kÙ(a.c).     called the expanded inner product rule with special case a.(bÙc) = (a.b)c - bÙ(a.c).
• (aÙb_k)i =  a.(b_ki)
• c₂.(aÙb) = (b.c₂).a = b.(c₂.a) =_( )= b . c₂ . a

    We will use ¿ in preference to . whenever possible here. a¿b = a.b " 1-vector a if b_<0>=0 so in many equations ¿ and . are interchangeable. However, only . is dual to Ù in the sense that a.(bi)   =   (aÙb)i    ;     ie. aÙ(bi)   =   (a.b)i ; or equivalently a.(b^*) = (aÙb)^* ; aÙ(b^*) = (a.b)^* ;
    a.b = (aÙ(bi^-1))i    ;     aÙb = (a.(bi^-1))i    for