A Factoring Algorithm for Clifford Appell
Decomposition
René Schott∗, G. Stacey Staples†
Abstract
Considered here is the problem of blade factorization in Clifford algebras of arbitrary quadratic form, and the role played by blade factorization
in construction of Clifford Appell systems. In particular, a homogeneous
multivector of grade k is a Clifford k-blade if it can be factored into the
Clifford (geometric) product of k anisotropic vectors that are orthogonal with respect to the quadratic form Q on the underlying vector space.
A representation-theoretic view of blade factorization algorithms is presented herein based on operator representations in Clifford algebras. As
an application, the role of blade factorization in the construction of a full
Clifford Appell system from a given blade is discussed in detail.
AMS Subj. Classifications: 15A66, 81R05, 20C40, 60B99
Keywords: Clifford algebras, factoring, algorithms, Appell decomposition,
operator calculus, quantum probability, Fock space, fermion
1
Introduction
Appell systems can be interpreted as polynomial solutions of generalized heat
equations, and in probability theory they can be used to obtain non-central limit
theorems. Their analogues have been defined on Lie groups [5], the Schrödinger
algebra [4], and quantum groups [3].
Appell systems on Clifford algebras are natural objects of interest for constructing solutions of Clifford evolution equations. The current authors first
defined general Appell systems within a Clifford algebra of arbitrary signature
in [8], and extended that work by considered left and right invertible Appell
systems on the n-particle fermion algebra [12]. The reader is directed to the
book [13] for a summary of the authors’ earlier work on Clifford Appell systems
and operator calculus.
∗ Université de Lorraine, IECN, UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France, email:
schott@loria.fr
† Department of Mathematics and Statistics, Southern Illinois University Edwardsville,
Edwardsville, IL 62026-1653,USA, email: sstaple@siue.edu
1
Other contributions to the study of Appell systems in the context of Clifford
algebras include the works of Eelbode [2] and Güerlebeck [6].
Historically, a homogeneous multivector of grade k is considered to be a kblade if it can be factored into an exterior product of k vectors. The vectors
occurring in the factorization are not required to be orthogonal, as the exterior multiplication naturally excludes non-orthogonal components. Fast blade
factorizations are known to exist in this context [1].
Of specific interest here is the separate problem of blade factorization in Clifford algebras of an arbitrary quadratic form along with the role played by such
a process in constructing Clifford Appell systems. A homogeneous multivector
of grade k is considered a Clifford k-blade if it can be factored into the Clifford
(geometric) product of k anisotropic vectors that are orthogonal with respect to
the quadratic form Q on the underlying vector space, V . Note that while every
Clifford k-blade is a blade in the traditional (exterior algebra) sense, not every
exterior algebra blade is a Clifford blade.
Presented herein is the development of a novel blade factorization algorithm
based on representations of induced maps. This algorithm makes use of operator
representations of the orthogonal group OQ (V ) to yield not only a factorization
of the given blade, but to establish simultaneously an anisotropic, Q-orthogonal
basis for the blade’s orthogonal complement in V . The algorithm thereby generates a full Appell decomposition of the Clifford algebra C`Q (V ).
The examples computed herein were generated with the corresponding author’s CliffMath11‘ package for Mathematica. The package, available for
download at http://www.siue.edu/~sstaple, facilitates computations in Clifford algebras of arbitrary signature C`p,q ' C`Q (V ) with canonical Q-orthonormal
generators {ei : 1 ≤ i ≤ p + q}.
2
Clifford Operator Calculus & Appell systems
Let V be an n-dimensional vector space over R equipped with a nondegenerate
quadratic form Q. Associate with Q the symmetric bilinear form
1
[Q(x + y) − Q(x) − Q(y)] ,
2
Vk
V by
and extend to simple k-vectors in
hx, yiQ =
(2.1)
hx1 ∧ x2 ∧ · · · ∧ xk , y1 ∧ y2 ∧ · · · ∧ yk iQ = det hxi , yj iQ .
(2.2)
Vk
V
This inner product extends linearly to all of
V and by orthogonality to V .
Given a product of Q-orthogonal vectors, u = e1 · · · ek , the reversion of u is
defined as
ũ := ek · · · e1 = (−1)k(k−1)/2 e1 · · · ek = (−1)k(k−1)/2 u.
(2.3)
Reversion extends linearly to an involution on C`Q (V ). The grade involution is
defined by linear extension of
û = e1\
· · · ek := (−1)k e1 · · · ek = (−1)k u.
2
(2.4)
The Q-inner product and exterior product extend to C`Q (V ) via the canonical vector space isomorphism. The left contraction operator is defined by (cf.
[7, Chapter 14])
xyy = hx, yiQ ∀x, y ∈ V ;
(2.5)
^
xy(u ∧ v) = (xyu) ∧ v + û ∧ (xyv), ∀u, v ∈
V, x ∈ V ;
^
(u ∧ v)yw = uy(vyw), ∀u, v, w ∈
V.
(2.6)
(2.7)
In particular, left and right contractions are dual to the exterior product and
satisfy the following:
huyv, wiQ = hv, ũ ∧ wiQ ,
(2.8)
huxv, wiQ = hu, w ∧ ṽiQ .
(2.9)
The Clifford algebra C`Q (V ) is the real algebra obtained from associative
linear extension of the geometric vector product
x y := hx, yiQ + x ∧ y, ∀x, y ∈ V.
(2.10)
Given an arbitrary Q-orthogonal basis {vi : 1 ≤ i ≤ n} for V , multi-index
notation for canonical basis blades is adopted in the following manner. Denote
the n-set {1, . . . , n} by [n], and denote the associated power set by 2[n] . The
ordered product of basis vectors (i.e., algebra generators) is then conveniently
denoted by
Y
vi = vI ,
(2.11)
i∈I
for any subset I ⊆ [n], also denoted I ∈ 2[n] .
Give a nondegenerate quadratic form Q, define the Q-seminorm of x ∈ V
by
kxkQ = |hx, xiQ |1/2 .
(2.12)
A vector x is said to be anisotropic if kxkQ 6= 0. A set S of Q-orthogonal vectors
is said to be Q-orthonormal if kxkQ = 1 for all x ∈ S.
Note that since Q is non-degenerate, all vectors of a Q-orthogonal basis
for V must be anisotropic. Given a collection of Q-orthogonal vectors {xi }, a
Q-orthonormal basis {ui : 1 ≤ i ≤ n} for V is obtained by defining
ui :=
xi
,
kxi kQ
(2.13)
for each i = 1, . . . , n. In particular, for each i = 1, . . . , n,
ui 2 = hui , ui iQ =
hxi , xi iQ
= ±1.
|hxi , xi iQ |
These vectors then generate the Clifford algebra C`Q (V ).
3
(2.14)
For convenience, the Q-orthonormal basis {ei : 1 ≤ i ≤ n} for V is fixed and
always available. Mathematica examples found in later sections are computed
relative to this fixed basis.
These products of generators are referred to as basis blades for the algebra.
The grade of a basis blade is defined to be the cardinality of its multi-index. An
arbitrary element u ∈ C`Q (V ) has a canonical basis blade decomposition of the
form
X
u=
uI vI ,
(2.15)
I⊆[n]
where uI ∈ R for each multi-index
X I. The grade-k part of u ∈ C`Q (V ) is then
naturally defined by huik :=
uI vI . It is now evident that C`Q (V ) has a
|I|=k
canonical vector space decomposition of the form
C`Q (V ) =
n
M
hC`Q (V )ik .
(2.16)
k=0
An arbitrary element u ∈ C`Q (V ) is said to be homogeneous of grade k if
huik 6= 0 and hui` = 0 for all ` 6= k. As the degree of a polynomial refers
to the maximal exponent appearing in terms of the polynomial, an arbitrary
multivector u ∈ C`Q (V ) is said to be heterogeneous of grade k if huik 6= 0 and
hui` = 0 for ` > k.
A general Clifford element u is said to be anisotropic if uũ 6= 0. In this case,
u is invertible, and
ũ
.
(2.17)
u−1 =
uũ
A multivector that can be factored into a Clifford product of vectors is
referred to as a versor. A versor that can be factored into a product of k-vectors
is a k-versor. In general, versors are not homogeneous.
Definition 2.1. A homogeneous, grade-k multivector u ∈ C`Q (V ) is said to be
a Clifford k-blade if and only if
i. u is anisotropic, and
ii. u can be written as a Clifford product of vectors satisfying
u=
k
Y
x` = x1 ∧ · · · ∧ xk
(2.18)
`=1
for {xi : 1 ≤ i ≤ k} ⊂ V . Note that this equality holds if and only if the
vectors are Q-orthogonal.
Note that any linearly independent collection {v1 , . . . , vk } of anisotropic
vectors from V generates a k-blade u ∈ C`Q (V ) by
u = hv1 · · · vk ik = v1 ∧ · · · ∧ vk .
4
(2.19)
In particular, the grade-k part of a k-versor is a k-blade.
The term blade factorization refers to any process by which the constituent
vectors of a k-blade are recovered. Such factorizations are obviously not unique.
Before constructing Appell systems in Clifford algebras, an operator calculus
for Clifford algebras must be discussed. Within the context of this calculus, the
construction of Appell systems will be natural.
In the authors’ earlier work, the motivation for development of Clifford operator calculus was based on polynomial operator calculus [8, 11]. To begin,
raising and lowering operators were defined naturally in terms of polynomial
differentiation and integration operators on Clifford multivectors regarded as
polynomials in anticommuting variables.
The Clifford differentiation operator ∂/∂x is defined for anisotropic x ∈ V
by linear extension of
∂
u = xyu.
(2.20)
∂x
Similarly, the Clifford integrals are defined by
Z
dx = x ,
(2.21)
Z Z
Z
dx dy = x dy = y ∧ x.
(2.22)
These polynomial operators induce combinatorial raising and lowering operators by which Clifford monomials (blades) are “raised” from grade k to grade
k + 1 or “lowered” from grade k to grade k − 1. These raising and lowering
operators can also be regarded as fermion creation and annihilation operators
in the sense of quantum mechanics.
Significantly, given a Q-orthogonal basis {xj : 1 ≤ j ≤ n} of V , the exterior
product and left contraction act as combinatorial raising and lowering operators
on multi-indices of blades in C`Q (V ):
(
ϑ({j}, I)xI∪{j} if j ∈
/ I,
xj ∧ xI = hxj xI i|I|+1 =
(2.23)
0
otherwise;
and
(
xj yxI = hxj xI i|I|−1 =
ϑ({j}, I)xI\{j}
0
if j ∈ I,
otherwise.
(2.24)
Here, the product signature map ϑ : 2[n] × 2[n] → {±1} is defined naturally
in terms of a counting measure on finite sets. For fixed positive integer j, define
the map µj : 2[n] → N0 by
µj (I) := |{i ∈ I : i > j}|.
In other words, µj (I) is the counting measure of the set {i ∈ I : i > j}.
5
(2.25)
For two multi-indices I, J ∈ 2[n] , defining
ϑ(I, J) = (−1)(
P
j∈J
µj (I))
Y
hx` , x` iQ
(2.26)
`∈I∩J
gives xI xJ = ϑ(I, J)xI4J , where I4J = (I ∪ J) \ (I ∩ J) denotes the setsymmetric difference of I and J.
Definition 2.2. Let x be an anisotropic vector in C`Q (V ), and define the xlowering operator Λx on C`Q (V ) by
Λx u =
∂
u = xyu
∂x
(2.27)
for any u ∈ C`Q (V ).
The (left) x-lowering operator Λx is correctly regarded as an operator taking
elements of grade k to elements of grade k − 1 for k = 1, . . . , n.
Definition 2.3. Let x be an anisotropic vector in C`Q (V ) and define the corresponding x-raising operator Ξx on C`Q (V ) by
Z
Ξx u = u dx = x ∧ u
(2.28)
for any u ∈ C`Q (V ).
The role of raising and lowering operators in the Clifford (geometric) product
is made explicit by considering the left regular representation of multiplication
by a vector x. Specifically, this is the operator sum (Ξx ⊕ Λx ), as seen by
xu = (Ξx ⊕ Λx )u
(2.29)
for u ∈ C`Q (V ).
The relationship between the generalized raising and lowering operators is
made clear by the next lemma.
Lemma 2.4. For fixed anisotropic vector x in C`Q (V ), the operators Ξx and Λx
are dual to each other with respect to the inner product h·, ·iQ ; i.e., hΛx u, wiQ =
hu, Ξx wiQ for all u, w ∈ C`Q (V ).
Proof. The result follows immediately from (2.8) and the definitions of the lowering and raising operators.
Defining the composition of lowering (differential) operators as multivector
left contraction operators also makes sense. Specifically, for fixed grade-k basis
blade xJ and arbitrary basis blade xI ,
∂
∂
···
xI = xj1 y(· · ·y(xjk yxI )) := xJ yxI .
∂xj1
∂xjk
6
(2.30)
As a consequence,
(
ϑ(J, I)xI\J
xJ yxI :=
0
if J ⊆ I,
otherwise.
(2.31)
This multivector contraction operator extends linearly to all of C`Q (V ). This
contraction operator is correctly regarded as a k th order lowering operator
ΛxJ xI = ϑ(J, I)xI\J ∈ hC`Q (V )i|I|−|J| .
(2.32)
One defines multivector right contraction operators in similar fashion.
Due to associativity, the exterior product already has a natural generalization; i.e.,
xI ∧ xJ = xi1 ∧ · · · ∧ xi|I| ∧ xj1 ∧ · · · ∧ xj|J| .
(2.33)
Consequently,
(
xI ∧ xJ =
ϑ(I, J) xI∪J
0
if I ∩ J = ∅,
otherwise.
(2.34)
Note that the general blade product xI xJ has the operator calculus formulation
xI xJ = Λxi1 + Ξxi1 ◦ · · · ◦ Λxi|I| + Ξxi|I| eJ .
(2.35)
A more general analogue to the number operator of quantum mechanics is
the grade operator defined on blades by
Γ(u1 ∧ · · · ∧ uk ) = k u1 ∧ · · · ∧ uk .
(2.36)
Lemma 2.5. For any anisotropic vector x ∈ C`Q (V ), the corresponding xlowering and x-raising operators are nilpotent of index 2. That is,
Λx 2 := Λx ◦ Λx = 0,
2
Ξx := Ξx ◦ Ξx = 0.
(2.37)
(2.38)
Proof. The result follows immediately from the properties of left contractions
and exterior products.
2.1
Blade Appell systems
Generally speaking, for an operator X on an algebra A, one sets
Zn = {ψ ∈ A : X n+1 ψ = 0}
for n ≥ 0, and defines an X -Appell system as a sequence of nonzero functions
{ψ0 , ψ1 , . . . , ψn , . . .} satisfying
i. ψn ∈ Zn , ∀n ≥ 0, and
7
ii. X ψn = ψn−1 , for n ≥ 1.
The system of embeddings Z0 ⊂ Z1 ⊂ Z2 ⊂ · · · is referred to as a canonical
X -Appell system decomposition.
In the Clifford algebra context, combinatorial raising and lowering operators
are natural choices for constructing Appell systems. These operators, which
correspond to the creation and annihilation operators of quantum mechanics,
map blades of grade k to blades of grade k + 1 and k − 1, respectively.
Appell systems are particularly useful as solutions of evolution equations.
One straightforward example is
∂t u = Λu,
(2.39)
where u = u(t) ∈ C`Q (V ) and Λ is an operator acting as generalized differentiation, or combinatorial lowering. Similarly, one can have an equation of the
form
∂t u = Ξu,
(2.40)
where Ξ is a generalized integral, or combinatorial raising. Considering discrete
processes of sums of raising and lowering operators gives
∂t u = (Λ + Ξ)u,
(2.41)
which can be regarded as a random walk on a directed hypercube [9, 10].
Definition 2.6. A collection {ψk : 0 ≤ k ≤ n} ⊂ C`Q (V ) is said to be a blade
system if the following conditions are satisfied:
i. ψ0 is a nonzero scalar and ψ1 is an anisotropic vector,
ii. ψk is a k-blade for 2 ≤ k ≤ n, and
iii. ψk−1 yψk ∈ V is an anisotropic vector for each 2 ≤ k ≤ n.
Given a blade system {ψk }, it is possible to define raising and lowering
operators Ξ and Λ, nilpotent of index 2 satisfying the following:
i. Λψj = ±ψj−1 for each 1 ≤ j ≤ n,
ii. Ξψj−1 = ±ψj for each 1 ≤ j ≤ n, and
iii. Λψ0 = Ξψn = 0.
Perhaps the simplest way to construct these operators is to define x1 := ψ1
and
xk := ψk−1 yψk , k = 2, . . . n.
(2.42)
V
V
k
k−1
For each k = 1, . . . , n, the k th lowering operator Λk :
Rn →
Rn is
defined by
(
xk yu if u is a k-blade,
Λk u :=
(2.43)
0
otherwise.
8
The lowering operator Λ is then determined by
Λ :=
n
M
Λk .
(2.44)
k=1
Vk−1 n
Vk n
For each k = 1, . . . , n, the k th lowering operator Ξk :
R →
R is
defined by
(
xk ∧ u if u is a (k − 1)-blade,
(2.45)
Ξk u :=
0
otherwise,
with the convention that the exterior product of a scalar with a vector is taken
to be scalar multiplication.
The raising operator Ξ is then determined by
Ξ :=
n
M
Ξk .
(2.46)
k=1
The blade system {ψk } taken together with the operators Λ and Ξ now
constitute an Appell system, referred to herein as a blade Appell system.
Definition 2.7. A blade Appell system on C`Q (V ) is defined as a triple
Ψ({ψk }, Λ, Ξ), where Λ and Ξ are lowering and raising operators, respectively,
such that
i. ψ0 is a nonzero scalar and ψ1 is an anisotropic vector,
ii. ψk is a k-blade for 2 ≤ k ≤ n,
iii. ψk−1 yψk ∈ V is an anisotropic vector for each 2 ≤ k ≤ n,
iv. Λψj = ±ψj−1 for each 1 ≤ j ≤ n,
v. Ξψj−1 = ±ψj for each 1 ≤ j ≤ n, and
vi. Λψ0 = 0 = Ξψn .
The problem at hand is twofold:
1. Given a homogeneous multivector u of grade k, determine whether u is a
blade.
2. Given a k-blade, u, construct a blade Appell system Ψ({ψj }, Λ, Ξ) satisfying u = ψk .
9
3
Endomorphisms of V and C`Q (V )
For any homogeneous grade-k multivector u ∈ C`Q (V ) satisfying 0 6= uũ ∈ R,
define the mapping ϕu : C`Q (V ) → C`Q (V ) by
ϕu (x) := u x
ũ
= uxu−1 .
uũ
(3.1)
For a fixed blade u, the linear map x 7→ uxu−1 is an endomorphism on
C`Q (V ) referred to as the conjugation of x by the blade u. More generally,
such a mapping will be referred to as blade conjugation. When the blade u is
normalized, i.e., uũ = ±1, the map has the form x 7→ uxũ.
As illustrated by the next lemma, when u is a Clifford blade the restriction
Φu := ϕu V
of ϕu to V determines an endomorphism on V .
Definition 3.1. An endomorphism A on V is Q-orthogonal if
hA(x), A(y)iQ = hx, yiQ .
(3.2)
The collection of all Q-orthogonal transformations on V forms a group called
the orthogonal group of Q, denoted OQ (V ). Specifically, T ∈ OQ (V ) if and only
if for every x ∈ V , Q(T x) = Q(x).
Lemma 3.2. If a homogeneous, grade-k multivector u ∈ C`Q (V ) is a Clifford
blade, then
1. 0 6= uũ ∈ R, and
2. u x ũ ∈ V for all x ∈ V .
Proof. Suppose u = u1 · · · uk is homogeneous of grade-k. Then
uũ = u1 · · · uk uk · · · u1 =
k
Y
hui , ui iQ ∈ R.
(3.3)
i=1
Moreover, decomposing an arbitrary vector x ∈ V into components parallel and
orthogonal to uk ,
uk xuk = uk (xk + x⊥ )uk = hxk , uk iQ uk − huk , uk iQ x⊥
= αxk + βx⊥ ∈ V, (3.4)
for scalars α and β. Associative extension gives uxũ ∈ V .
When Φu is an endomorphism on V having eigenvalue λ, let Eλ denote the
corresponding eigenspace. A blade test is now given by the following theorem.
10
Theorem 3.3. A homogeneous, grade-k multivector u ∈ C`Q (V ) is a blade if
and only if Φu is an endomorphism on V with eigenvalues λ1 = (−1)k−1 and
λ2 = −λ1 = (−1)k such that
dim(Eλ1 ) = k,
dim(Eλ2 ) = n − k.
Proof. First, the requirement that Φu be an invertible linear transformation on
V follows from Lemma 3.2. Next, suppose u is a blade and write u = x1 · · · xk
for Q-orthogonal anisotropic vectors {xi : 1 ≤ i ≤ n}. It follows immediately
that
Φu (xi ) =
1
x1 · · · xk xi xk · · · x1
uũ
(−1)k−i Q(xi )(−1)i−1 Y
xi
Q(x` )
=
uũ
`6=i
k−1
=
(−1)
uũ
xi
Y
Q(x` )
1≤`≤k
=
(−1)k−1
xi uũ = (−1)k−1 xi . (3.5)
uũ
Hence, {xi : 1 ≤ i ≤ k} is a basis for the eigenspace Eλ1 .
Further, if v is Q-orthogonal to B, then
Φu (v) =
1
x1 · · · xk vxk · · · x1
uũ
Y
(−1)k
v
Q(x` )
=
uũ
1≤`≤k
=
(−1)k uũ
v
uũ
= (−1)k v. (3.6)
Hence, v is an eigenvector of Φu associated with eigenvalue λ2 = (−1)k . Since
v was arbitrarily chosen from the orthogonal complement of Eλ1 , it follows that
dim(Eλ2 ) = n − k.
Conversely, consider Φu ∈ End(V ) having the prescribed eigenspaces. Let
x ∈ V be an arbitrary anisotropic vector and let B denote a Q-orthogonal basis
of V containing x. Rewriting u relative to B, and observing that uxu−1 = x ⇔
ux = xu, consider the case k ∼
= 0 (mod 2). It becomes evident that ux = xu
only if each term ui in the expansion of u relative to B commutes with x. Writing
ui = b1 · · · bk for Q-orthogonal vectors b1 , . . . , bk , one sees
ui x = xui ⇔ b1 · · · bk x = xb1 · · · bk ⇔ xyui = 0.
11
(3.7)
Hence, when u is homogeneous of even grade, x is an eigenvector of the
transformation Φu corresponding to eigenvalue λ = 1 if and only if x is Qorthogonal to every term in the expansion of u.
Similar reasoning shows that when u is homogeneous of odd grade, x is an
eigenvector of the transformation Φu corresponding to eigenvalue λ = −1 if and
only if x is Q-orthogonal to every term in the expansion of u.
On the other hand, suppose x appears in each term of the expansion of
u. That is, every term ui in the expansion of u can be written in the form
ui = b1 · · · bk−1 x for some Q-orthogonal subset {b1 , . . . , bk−1 } of B. In this case,
k even implies k − 1 is odd so that
ui x = b1 · · · bk−1 x x = −xb1 · · · bk−1 x = −xui ,
(3.8)
which implies uxu−1 = −x. Hence, x is an eigenvector of Φu corresponding to
eigenvalue λ = −1.
Similar reasoning shows that when u is homogeneous of odd grade, x is
an eigenvector of the transformation Φu corresponding to eigenvalue λ = 1 if
and only if x is a factor of every term in the expansion of u. In other words,
u = xu0 = x ∧ u0 for some homogeneous u0 of grade k − 1.
It follows immediately that u is a blade when Φu has a k-dimensional
eigenspace associated with eigenvalue λ1 = (−1)k−1 .
Remark 3.4. The geometric significance of the conjugation Φu (x) = uxu−1 is
well-known, particularly in the Euclidean signature, where conjugation by normalized blades corresponds to compositions of hyperplane reflections, yielding
reflections and rotations in Euclidean space.
Theorem 3.5 (Blade factorization). Given an arbitrary homogeneous, grade-k
multivector u ∈ C`Q (V ) satisfying the conditions of Theorem 3.3. It follows that
Φu is a linear transformation of rank n and that any linearly independent set
of eigenvectors {v1 , . . . , vk } associated with eigenvalue λ1 = (−1)k−1 give an
exterior factorization of the blade u; i.e.,
u=α
k
^
v` = α v[k]
(3.9)
`=1
for some scalar α. Moreover, there exists a Q-orthogonal basis {w1 , . . . , wk } of
Eλ1 such that u has the Clifford (geometric) factorization
u=β
k
Y
w` = β w[k]
`=1
for some scalar β.
Proof. Claim. The homomorphism Φu is an involution, i.e., Φu 2 = I.
12
(3.10)
Proof of claim. Letting g denote the grade of u, note that ũ = (−1)g(g−1)/2 ,
so that uxũ = (−1)g(g−1)/2 uxu = ũxu. Assuming for convenience that uũ = 1,
it follows that for arbitrary x ∈ C`Q (V ),
Φu (Φu (x)) = u(uxũ)ũ = u(ũxu)ũ = (uũ)x(uũ) = x.
(3.11)
Since Φu is an involution, the right-regular representation is diagonalizable
with eigenvalues ±1 by a standard result in linear algebra.
Suppose u = u1 · · · uk for Q-orthogonal vectors u1 , . . . , uk . Then, for each
i = 1, . . . , k,
Φu (ui ) = uui
=
ũ
uũ
(−1)k−i
hui , ui iQ (u1 · · · ui−1 )(ui+1 · · · uk uk · · · ui+1 )ui (ui−1 · · · u1 )
uũ
k
(−1)k−i (−1)i−1 Y
huj , uj iQ ui
=
uũ
j=1
= (−1)k−1
uũ
ui = (−1)k−1 ui . (3.12)
uũ
It follows that each factor ui of the blade u is in the eigenspace of Φu
corresponding to eigenvalue (−1)k−1 . Moreover, the basis eigenvectors B =
{v1 , . . . , vk } of Eλ1 are anisotropic. To see this, note that u = αv1 · · · vk and
suppose vi ∈ B is isotropic. Then,
Φu (vi ) = uvi
Q(vi )
ũ
=±
uũ
uũ
Y
Q(v` ) = 0,
(3.13)
1≤`6=i≤k
contradicting Φu (vi ) = (−1)k−1 vi .
Note that the eigenvectors of Φu corresponding to eigenvalue λ1 are not
necessarily Q-orthogonal. However, since the vectors are anisotropic, a Qorthogonal basis γu = {w1 , . . . , wk } of Eλ1 exists. Since the factors of u span
this same k-dimensional subspace of V , a suitable scalar α yields (3.10).
Lemma 3.6. The eigenspaces Eλ1 and Eλ2 are orthogonal with respect to the
quadratic form Q.
Proof. Begin by letting v ∈ Eλ1 and w ∈ Eλ2 . Then,
uvu−1 uwu−1 = λ1 λ2 vw = λ1 λ2 (hv, wiQ + v ∧ w) .
(3.14)
Letting α = hv, wiQ and noting that λ1 λ2 = −1, this implies
u(vw)u−1 = −α − v ∧ w.
(3.15)
On the other hand, noting that vw = α + v ∧ w, one also finds
u(vw)u−1 = u (α + v ∧ w) u−1 = α − v ∧ w.
Equality of (3.15) and (3.16) then implies α = 0.
13
(3.16)
Recalling the eigenvalues λ1 = (−1)k−1 and λ2 = (−1)k associated with
conjugation by a blade of grade k, the next result is immediate.
1
Lemma 3.7. Given a k-blade u, the operators (I + λ1 Φu ) and
2
1
(I − λ1 Φu ) act as projections onto Eλ1 and Eλ2 , respectively.
2
Proof. Recall that λ1 = (−1)k−1 and λ2 = (−1)k . If v ∈ Eλ1 , it follows immediately that
1
1
I + (−1)k−1 Φu (v) =
v + λ1 2 v = v.
(3.17)
2
2
Further, v ∈ Eλ2 implies
1
1
I + (−1)k Φu (v) =
v + λ2 2 v = v.
2
2
(3.18)
The result then follows from Q-orthogonality of Eλ1 and Eλ2 .
For any linear transformation X ∈ L(V ), let ker(X) denote the kernel (or
null space) of X.
Corollary 3.8. Let u ∈ C`Q (V ) be a Clifford k-blade. Then, for any Qorthogonal basis {x1 , . . . , xk } of ker (I − λ1 Φu ), ∃α ∈ R such that
u=α
k
Y
xj .
j=1
Definition 3.9. A collection B = {u1 , . . . , uk } of anisotropic vectors in V is
said to be Q-orthogonalizable if there exists a permutation σ ∈ Sk such that
defining
j−1
X
huσ(j) , w` i
w` , ∀1 ≤ j ≤ k
(3.19)
wj := uσ(j) −
hw` , w` i
`=1
yields an anisotropic Q-orthogonal collection {w1 , . . . , wk }.
Example 3.10. Applying Gram-Schmidt orthogonalization to the collection
B = {e2 , e1 + e2 + e3 , e1 } in C`1,2 results in the collection
{e2 , e1 + e3 , −e3 }
which is not Q-orthogonal. In fact, u2 is isotropic. However, by permuting
the elements of B and applying Gram-Schmidt orthogonalization to the ordered
collection {e1 , e2 , e1 + e2 + e3 }, one obtains the anisotropic, Q-orthogonal collection
{e1 , e2 , e3 }.
Remark 3.11. It is not difficult to see that when Q corresponds to positive
definite or negative definite spaces, every linearly-independent set of vectors is
orthogonalizable.
14
Rather than computing the kernel of the operator I − λ1 Φu to obtain
a blade factorization, one can simply consider the row space of the rightregular representation of (I − λ2 Φu )/2, provided some k-subset of the rows
is Q-orthogonalizable.
Corollary 3.12. If applying Gram-Schmidt orthogonalization to the rows of
the right-regular representation of I − λ2 Φu yields a collection {w1 , . . . , wk } of
nonzero anisotropic vectors, then
u=α
k
Y
wk
i=1
for some scalar, α.
Proof. Note that ker(I − λ1 Φu ) = Im(I − λ2 Φu ).
Realizations of the endomorphisms of V considered in the examples here are
developed as n × n real matrices. The dual of an endomorphism A ∈ End(V ),
denoted A∗ , is therefore correctly regarded as the matrix transpose of A.
When A is regarded as an endomorphism in the general sense, the argument
of A will be made explicit, as in A(x). On the other hand, use of the matrix
representation of A will be indicated by the right action of the matrix; i.e., xA.
Given a Q-orthogonal basis {ui : 1 ≤ i ≤ n} for V , a right-regular matrix
representation of A ∈ End(V ) is determined by
Ai j = hui A, uj iQ = hA(ui ), uj iQ .
(3.20)
The relationships among Q, the Q-inner product, and the right-regular representation of Q are understood by
Q(x) = hx, xiQ = xQx∗ .
(3.21)
Example 3.13. The following example is computed in C`3,4 using Mathematica
and the CliffMath11‘ package. a randomly-generated 4-blade u ∈ C`3,4 is
depicted in (3.22). The right-regular representation of the corresponding blade
conjugation operator is given by (3.23).
u = −24e{1,2,3,4} − 36e{1,2,3,5} − 36e{1,2,3,6} − 72e{1,2,3,7} − 24e{1,2,4,7}
+ 36e{1,2,5,7} + 36e{1,2,6,7} + 120e{1,3,4,5} + 144e{1,3,4,6} + 360e{1,3,4,7}
− 36e{1,3,5,6} − 180e{1,3,5,7} − 108e{1,3,6,7} + 120e{1,4,5,7} + 144e{1,4,6,7}
− 36e{1,5,6,7} + 108e{2,3,4,5} + 108e{2,3,4,6} + 312e{2,3,4,7} − 144e{2,3,5,7}
− 144e{2,3,6,7} + 108e{2,4,5,7} + 108e{2,4,6,7} − 108e{3,4,5,6} − 60e{3,4,5,7}
+ 252e{3,4,6,7} − 144e{3,5,6,7} + 108e{4,5,6,7} . (3.22)
15

68
− 127


 178
−
 127

 19

 127

 45
Φ=

 127

 55
−
 127

 123

 127


19
−
127
178
−
127
83
−
381
46
127
2
127
92
−
381
124
127
46
−
127
19
127
46
127
179
−
127
63
−
127
50
−
127
96
−
127
52
127
45
−
127
2
−
127
63
127
58
−
127
85
127
87
127
63
−
127
55
127
92
381
50
127
85
127
281
381
64
−
127
50
−
127
123
−
127
124
−
127
96
127
87
127
64
−
127
187
127
96
−
127

19
127 


46 

127 

52 

−
127 

63 

−

127 

50 

−
127 

96 

−
127 


75
−
127
(3.23)
In (3.24), the right-regular representation of the operator (I − Φu )/2, which
acts by projection onto the eigenspace of λ1 = −1, is constructed. An orthonormal basis ((3.25) - (3.28)) for the row space is then obtained by applying
Q-Gram-Schmidt orthonormalization directly to the rows of the matrix.

195
 254


 89

 127

 19
−
 254

 45
(I − Φ)/2 = 
−
 254

 55

 254

 123
−
 254


19
254
r
v1 =
89
127
232
381
23
−
127
1
−
127
46
381
62
−
127
23
127
19
−
254
23
−
127
153
127
63
254
25
127
48
127
26
−
127
45
254
1
127
63
−
254
185
254
85
−
254
87
−
254
63
254
55
−
254
46
−
381
25
−
127
85
−
254
50
381
32
127
25
127
123
254
62
127
48
−
127
87
−
254
32
127
30
−
127
48
127

19
−
254 


23 

−
127 

26 

127 

63 
 (3.24)

254 

25 

127 

48 

127 


101
127
r
r
19 e{3}
195
2
15
e{1} + 89
e{2} − √
e{4}
+3
254
24765
3302
49530
r
r
19 e{7}
5
3
− 11
e{5} + 41
e{6} − √
9906
16510
49530
(3.25)
16
r
v2 = −
2
11
e{2} −
65
3
1
v3 =
3
r
2
e{3} −
65
r
10
e{4} +
13
r
5
e{5}
26
r
3 e{6}
11 2
+ √
−
e{7}
3 65
130
(3.26)
r
r
3 e{5}
3 e{6}
11 e{7}
29
2
e{3} +
e{4} − √
− √
+ √
2
29
58
58
3 58
r
r
3 e{5}
3 e{6}
2
2
e{4} − √
− √
−3
e{7}
v4 =
29
29
58
58
(3.27)
(3.28)
In (3.29), the Clifford product of the basis vectors of (3.25)-(3.28) is computed. The 4-blade
√ u of (3.22) is then recovered by scaling this product by the
factor α = −24 381; i.e., u = αΥ.
r
r
r
e{1,2,3,4}
1
1
3
3
3
+
e{1,2,3,5} +
e{1,2,3,6} +
e{1,2,3,7}
Υ=− √
2 127
2 127
127
381
r
r
e{1,2,4,7}
5 e{1,3,4,5}
3
3
1
1
+ √
−
e{1,2,5,7} −
e{1,2,6,7} − √
2 127
2 127
381
381
r
r
r
1
3
3
3
−2
e{1,3,4,6} − 5
e{1,3,4,7} +
e{1,3,5,6}
127
127
2 127
r
r
5 e{1,4,5,7}
5
3
3
3
+
e{1,3,5,7} +
e{1,3,6,7} − √
2 127
2 127
381
r
r
r
r
3
1
3
3
3
3
3
−2
e{1,4,6,7} +
e{1,5,6,7} −
e{2,3,4,5} −
e{2,3,4,6}
127
2 127
2 127
2 127
r
r
r
13 e{2,3,4,7}
3
3
3
3
√
e{2,3,5,7} + 2
e{2,3,6,7} −
e{2,4,5,7}
−
+2
127
127
2 127
381
r
r
r
5 e{3,4,5,7}
3
7
3
3
3
3
√
−
−
e{2,4,6,7} +
e{3,4,5,6} +
e{3,4,6,7}
2 127
2 127
2 127
2 381
r
r
3
3
3
+2
e{3,5,6,7} −
e{4,5,6,7} (3.29)
127
2 127
3.1
Appell Systems from Blades
Given a Clifford k-blade, u, it is now possible to construct a blade Appell system
satisfying ψk = u.
1. Set ψk := u.
2. Compute an ordered Q-orthogonalizable basis B = {v1 , . . . , vk } for the
null space of I − λ1 Φu .
17
3. Apply Q-Gram Schmidt orthonormalization to the vectors of B to obtain
an ordered Q-orthonormal basis B 0 = {w1 , . . . , wk }.
Qk
4. Compute the scalar α such that u = i=1 wi .
5. Set ψj−1 := Λwj ψj for j = 1, . . . , k.
6. Compute an ordered Q-orthogonalizable basis B⊥ = {vn+1 , . . . , vn } for
the null space of I − λ2 Φu .
7. Apply Q-Gram Schmidt orthonormalization to the vectors of B⊥ to obtain
0
an ordered Q-orthonormal basis B⊥
= {wn+1 , . . . , wn }.
8. Set ψj+1 := Ξwj ψj for j = k, . . . , n.
Ln
Ln
9. Define Λ := `=1 Λw` and Ξ := `=1 Ξw` , as before.
Given any anisotropic Q-orthogonal basis {ei } for V , the pseudoscalar for
C`Q (V ) is the n-blade e[n] . For a k-blade u, let u? denote the dual of u, defined
implicitly by u? u = e[n] . Note that u? is a blade of grade n − k and satisfies
u? u = e[n] = (−1)k(n−k) uu? .
(3.30)
Example 3.14. In (3.31), the right-regular representation of the operator
(I + Φu )/2, which acts by projection onto the eigenspace of λ2 = 1, is constructed. An orthonormal basis, (3.32)-(3.34), for the row space is then obtained by applying Q-Gram-Schmidt orthonormalization directly to the rows of
the matrix.

59
 254


 89
−
 127

 19

 254

 45
(I + Φ)/2 = 

 254

 55
−
 254

 123

 254


19
−
254
89
−
127
149
381
23
127
1
127
46
−
381
62
127
23
−
127
19
254
23
127
26
−
127
63
−
254
25
−
127
48
−
127
26
127
45
−
254
1
−
127
63
254
69
254
85
254
87
254
63
−
254
18
55
254
46
381
25
127
85
254
331
381
32
−
127
25
−
127
123
−
254
62
−
127
48
127
87
254
32
−
127
157
127
48
−
127

19
254 


23 

127 

26 

−
127 

63 
 (3.31)
−

254 

25 

−
127 

48 

−
127 


26
127
r
v6 =
59
e{1} − 89
254
r
r
45 e{4}
55 e{5}
19 e{3}
2
−√
+√
e{2} + √
7493
14986
14986
14986
123 e{6}
19 e{7}
− √
+√
14986
14986
r
r
137 e{5}
3
e{4} + √
17995
53985
r
15
3
− 23
e{6} + 24
e{7}
3599
17995
r
r
21 e{5}
9 e{7}
9 e{3}
2
5
+6
e{4} + √
+
e{6} − √
.
v8 = − √
305
122
610
610
610
v7 = −
305
e{2} + 24
177
3
e{3} − 32
17995
r
(3.32)
(3.33)
(3.34)
In (3.35), the Clifford product of the basis vectors (3.32)-(3.34) is computed.
The 3-blade u? is then recovered by scaling this product by the factor α? =
1
1
√
; i.e., u? = α? Υ? = − Υ.
α
24 381
r
r
r
5 e{1,2,6}
3
3
3
7
3
Υ? =
e{1,2,3} − 2
e{1,2,4} −
e{1,2,5} − √
2 127
127
2 127
2 381
r
r
r
r
3
3
3
3
3
3
3
+
e{1,2,7} +
e{1,3,5} −
e{1,3,6} − 2
e{1,4,5}
2 127
2 127
2 127
127
r
r
r
13 e{1,5,6}
3
3
3
3
3
e{1,4,6} + √
−
e{1,5,7} +
e{1,6,7}
+2
127
2 127
2 127
381
r
r
r
5 e{2,3,6}
1
3
3
3
3
−
+
e{2,3,4} − 2
e{2,3,5} + √
e{2,4,5}
2 127
127
2
127
381
r
r
r
r
3
1
3
3
3
5
−
e{2,4,6} +
e{2,4,7} − 5
e{2,5,6} + 2
e{2,5,7}
2 127
2 127
127
127
r
r
5 e{2,6,7}
e{3,5,6}
1
3
1
3
+
− √
e{3,4,5} −
e{3,4,6} − √
2 127
2 127
381
381
r
r
r
e{5,6,7}
3
1
3
1
3
−
e{4,5,6} +
e{4,5,7} −
e{4,6,7} − √
(3.35)
127
2 127
2 127
381
Example 3.15. Based on Examples 3.13 and 3.14, it is now possible to construct a full blade Appell system Ψ = ({ψk }, Λ, Ξ) in C`3,4 satisfying ψ4 = u.
The blade system is seen in (3.36)-(3.38) and (3.39)-(3.41).
19
r
2
ψ3 := Λψ4 = 6
58 e{1,2,3} + 12 e{1,2,4} − 18 e{1,2,5} − 18 e{1,2,6}
29
r
2
+6
22 e{1,2,7} − 246 e{1,3,4} + 79 e{1,3,5} + 21 e{1,3,6} − 132 e{1,3,7}
29
r
2
+6
−60 e{1,4,5} − 72 e{1,4,6} + 66 e{1,4,7} + 18 e{1,5,6}
29
r
2
+6
11 e{1,5,7} + 33 e{1,6,7} − 210 e{2,3,4} + 54 e{2,3,5} + 54 e{2,3,6}
29
r
2
+6
−124 e{2,3,7} − 54 e{2,4,5} − 54 e{2,4,6} + 54 e{2,4,7} + 18 e{2,5,7}
29
r
2
+6
18 e{2,6,7} + 57 e{3,4,5} − 153 e{3,4,6} + 48 e{3,4,7}
29
r
2
+6
54 e{3,5,6} − 46 e{3,5,7} + 78 e{3,6,7} − 54 e{4,5,6}
29
r
2
+6
(3.36)
27 e{4,5,7} − 27 e{4,6,7} − 18 e{5,6,7}
29
ψ2 := Λψ3 = 4 18 e{1,2} + 66 e{1,3} + 90 e{1,4} − 45 e{1,5} − 27 e{1,6}
+ 4 66 e{1,7} + 62 e{2,3} + 78 e{2,4} − 36 e{2,5} − 36 e{2,6}
+ 4 62 e{2,7} − 24 e{3,4} + 23 e{3,5} − 39 e{3,6} + 15 e{4,5} − 63 e{4,6}
(3.37)
+ 4 24 e{4,7} + 36 e{5,6} − 23 e{5,7} + 39 e{6,7}
r
ψ1 := Λψ2 = −12
2
195 e{1} + 178 e{2} − 19 e{3} + 45 e{4} − 55 e{5}
65
r
2
− 12
123 e{6} − 19 e{7}
(3.38)
65
r
254
9 e{1,2,3,4,5} + 9 e{1,2,3,4,6} + 26 e{1,2,3,4,7}
59
254
+ 12
−12 e{1,2,3,5,7} − 12 e{1,2,3,6,7} + 9 e{1,2,4,5,7}
59
r
254
+ 12
9 e{1,2,4,6,7} − 9 e{1,3,4,5,6} − 5 e{1,3,4,5,7}
59
r
254
+ 12
21 e{1,3,4,6,7} − 12 e{1,3,5,6,7} + 9 e{1,4,5,6,7}
59
ψ5 := Ξψ4 = 12
r
20
(3.39)
r
ψ6 := Ξψ5 = 12
762
9 e{1,2,3,4,5,6} + 5 e{1,2,3,4,5,7} − 21 e{1,2,3,4,6,7}
305
r
762
12 e{1,2,3,5,6,7} − 9 e{1,2,4,5,6,7}
(3.40)
+ 12
305
√
ψ7 := Ξψ6 = −24 381 e{1,2,3,4,5,6,7} .
4
(3.41)
Conclusion
Aside from its role in constructing Appell systems, blade factorization has farreaching implications. For example, the ability to construct full Appell systems
from partial data obtained from a single multivector is related to problems of
interpolation and approximation using Clifford algebras.
Blade factorization also has applications as a form of “Clifford compression.”
In particular, a k-blade of an n-dimensional vector space V stored as a linear
combination of canonical basis blades can require up to nk scalar coefficients.
On the other hand, a suitable choice of basis reduces this to storing k vectors
of length n. For example, given a canonical basis for C`10,0 , the canonical
multivector expansion of an arbitrary 5-blade may have as many as 10
5 = 252
nonzero coefficients, while the right-regular representation of Φu is a 10 × 10
matrix. Reducing this to a 5-vector basis {x1 , . . . , x5 } for the eigenspace of
λ1 results in storing only 50 scalars plus the scaling factor α satisfying u =
αx1 · · · x5 . As the dimension increases, so do the savings.
References
[1] L. Dorst, D. Fontijne, Efficient Algorithms for Factorization and
Join of Blades, In Geometric Algebra Computing, E. Bayro-Corrochano,
G. Scheuermann, Eds., Springer, London, 2010, pp. 457–476.
[2] D. Eelbode, Monogenic Appell sets as representations of the Heisenberg
algebra, Adv. Appl. Clifford Algebras, (2012), Online at http://www.
springerlink.com/content/0188-7009/. DOI 10.1007/s00006-012-0330-z
[3] P. Feinsilver, U. Franz, R. Schott, Duality and multiplicative processes on
quantum groups, J. Th. Prob., 10 (1997), 795-818.
[4] P. Feinsilver, J, Kocik, R. Schott, Representations of the Schrödinger algebra
and Appell systems, Progress of Phys., 52 (2004), 343–359.
[5] P. Feinsilver, R. Schott, Appell systems on Lie groups, J. Th. Prob., 5 (1992),
251–281.
21
[6] N. Güerlebeck, On Appell sets and the Fueter-Sce mapping, Adv. Appl.
Clifford Algebras, 19 (2009), 51–61.
[7] P. Lounesto, Clifford Algebras and Spinors, Cambridge University Press,
Cambridge, 2001.
[8] R. Schott, G.S. Staples, Operator calculus and Appell systems on Clifford
algebras, Int. J. of Pure and Appl. Math., 31 (2006), 427–446.
[9] R. Schott, G.S. Staples. Random walks in Clifford algebras of arbitrary
signature as walks on directed hypercubes, Markov Processes and Related
Fields, 14 (2008), 515–542.
[10] R. Schott, G.S. Staples. Dynamic random walks in Clifford algebras, Advances in Pure and Applied Mathematics, 1 (2010), 81–115.
[11] R. Schott, G.S. Staples. Operator homology and cohomology in Clifford
algebras, Cubo, A Mathematical Journal, 12 (2010), 299–326.
[12] R. Schott, G.S. Staples, Operator calculus and invertible Clifford Appell
systems: theory and application to the n-particle fermion algebra, Preprint,
http://www.loria.fr/~schott/staceyIDAQP11.pdf.
[13] R. Schott, G.S. Staples, Operator Calculus on Graphs (Theory and Applications in Computer Science), Imperial College Press, London, 2012.
22