glossary

algebra :

to form any algebra, we need elements and an associative product among them.

Euclidian space :

(This definition is for dimension 2 but this may be generalized for any dimension).

Consider the plane R x R = {(x,y) | x,y <- R}. Introduce a linear structure by addition (x1,y1)+(x2,y2)=(x1+x2,y1+y2) and by scaling µ(x,y) = (µx, µy), where µ <- R. The linear structure makes the plane R x R into a linear space R². Take a basis (e1,e2) of R², say e1=(1,0), e2=(0,1). Introduce the length of r = xe1 + ye2 as |r| = sqrt(x² + y²). The introduction of length makes R² into an Euclidian plane, also denoted by R². The basis vectors e1, e2 are unit vectors, |e1|=1, |e2|=1.

Related to length there is a scalar valued product a.b = a1b1 + a2b2 of two vectors

a = a1e1 + a2e2, b = b1e1 + b2e2 <- R². The scalar product is symetric, a.b = b.a.

Two vectors a and b are orthogonal if their scalar product vanishes, a.b = 0. The unit vectors e1 and e2 are orthogonal. (P. Lounesto)

Clifford product :

Next, we introduce an associative, but non commutative (nonsymetric) product for vectors in R². If the vector r is multiplied by itself or squared, r r = r², we require that the square of the vector equals the square of the length of the vector : r² = |r|².

In coordinate form, we introduce a product for vectors such that : (xe1 + ye2)² = x² + y²

Using the distributive law, without assuming commutativity, gives : x²e1² +y²e2² + xy(e1e2 + e2e1) = x² + y²

This is satisfied if the orthogonal unit vectors e1, e2 obey the multiplication rules

e1² = e2² = 1 wich correspond to |e1| = |e2| = 1

e1e2 = - e2e1 e1 ⊥ e2

Using associativity and anticommutativity we can calculate the square as (e1e2)² = -e1²e2² = -1. Since the square of the product e1e2 is negative, it folows that e1e2 is neither a scalar nor a vector. The product e1e2 is a new kind of quantity called a bivector, and represents the oriented plane area of the square with sides e1 and e2. Write for short e12 = e1e2

Define for two vectors a = a1e1 + a2e2, b = b1e1 + b2e2 the Clifford product (Geometric product)

ab = (a1b1 + a2b2) + (a1b2 - a2b1)e12, a sum of a scalar and a bivector. (P. Lounesto)

outer product : a ^ b = - b ^ a

Generalize the cross product to higher dimensions

The outer product of r vectors a1 ^ a2 ^ ...^ ar can be defined as the anti-symmetric part of the geometric product of the r vectors. It is called an r-blade, or a blade of grade r Ar.

Ar ^ Bs = <A B>r+s

inner product : a . b = b . a

Generalize the scalar product to higher dimentions

Ar . Bs = <A B>|r-s|

geometric product : a b = a . b + a ^ b

decomposes into symmetric and anti-symmetric parts.

inner and outer product, function of geometric product : (with A' = invol(A) )

a ^ A = 1/2(a A + A' a) A ^ a = 1/2(A a + a A')

a . A = 1/2(a A - A' a) A . a = 1/2(A a - a A')

commutator product :

A x B = 1/2(AB - BA)

A x (BC) = (A x B)C + B(A x C)

A x (B x C) = (A x B) x C + B x (A x C) Jacobi identity

Scalar :

A point (real number) - grade 0 - no geometric extent

Vector :

A directed length - grade 1 - extent in 1 direction

Bivector :

A directed area - grade 2 - extent in 2 directions

Trivector :

A directed volume - grade 3 - extent in 3 directions

oriented volume :

In Cln, the maximal grade of a blade is n.

In Cl3 the trivector is a pseudoscalar : j = e123, j² = -1, |j| = 1.

blade :

A scalar, a vector, or the outer product of any number of vectors A = a1 ^ a2 ^ ...^ ar <A>r = A

grade :

Blade dimension r (= subspace dimension) <A>r

multivector :

Linear combination of blades A = <A>0 + <A>1 + ... + <A>r + ... + <A>n

reversion :

rev(a1 ...ar) = ar ...a1

Reverse basis element order e0 -> e0, e1 -> e1, e2 -> e2, ..., e12 -> e21, e23 -> e32, ..., e123 -> e321

grade involution :

invol(a1 ...ar) = (−1)**r (a1 ...ar)

Change basis element direction e0 -> e0, e1 -> -e1, e2 -> -e2, ..., e12 -> e12, e23 -> e23, ..., e123 -> -e123

Clifford conjugation :

cj(A) = rev(invol(A) = invol(rev(A)

conjugation = reversion(involution) = involution(reversion)

parity :

A+ even if invol(A) = A

A- odd if invol(A) = -A

A+ A+ = A+ --> define even subalgebra

dual :

dual(A) = A / j (j = e12..n, Cln)

dual(<A>r) = <A>n-r

(a · A) j = a ^ (A j)

(a ^ A) j = a · (A j)

in Cl3

pseudoscalar : j = e12..n, dual(j) = e0

translation :

Transformation wich preserve distances, direction and orientation.

reflection :

Transformation wich preserves distances and leaves a blade, the blade of reflection, unchanged.

rotation :

Transformation wich preserves distances and leaves a line, the axis of rotation, unchanged.

meet :

Intersection of vector subspaces.

A V B = dual(A) . B grade(A) + grade(B) >= n

projection :

PAr (x) = (x · Ar)/Ar .

rejection :

P⊥Ar (x) = (x ^ Ar)/Ar .

orthogonal decomposition :

x = PAr (x) + P⊥Ar (x)

automorphism :

Invertible linear maping that preserve geometric products : f(A B) = f(A) f(B)

anti-automorphism :

Linear mapping that reverse the order of geometric products : g(A B) = g(B) g(A)

The main anti-automorphism of vector algebra is called reversion : rev (a1 ...ar) = ar ...a1.

involution :

An involution h is an invertible linear mapping whose composition with itself is the identity map : h(h(A)) = A

The main involution of geometric algebra is called grade involution : invol (a1 ...ar) = (−1)**r (a1...ar).

magnitude :

value of a scalar, length of a vector, area of a bivector, volume of a trivector, the sum of all these in mixed multivector.

Extended from vectors to any multivector A by |A| = sqrt( sum (|Ai|,0,n)) where |Ai| = sqrt(Ai . Ai) and Ai = <A>i

scalar product :

Extended from vectors to any multivector A and B by <A rev(B)> = <B rev(A)> = sum (<Ai rev(Bi)>,0,n)

where <...> = <...>0 denote the scalar part.

linear independance :

if a1 ^ a2 ^ ...^ ar != 0

basis (frame) :

If a blade Ar admits the decomposition Ar = a1 ^ a2 ^ . . . ^ ar , the set of vectors {ai ; i = 1, . . . , r} is said to be a frame (or basis) for Ar (pseudoscalar).

A dual frame {ri} is defined by the equations ri · aj = δij.

If Ar is invertible, these equations can be solved for the ri, with the result

ri = (−1)**(i+1) (a1 ^ ... ^ai-1 ^ ai+1 ^ ... ^ ar)/Ar