A Symbolic Algorithm for Computation of Non-degenerate Clifford Algebra Matrix Representations

TL;DR

This paper presents an algorithmic method to construct matrix representations of non-degenerate Clifford algebras, facilitating automated proof checking and inverse computation of multivectors.

Contribution

It introduces a transparent, algorithmic approach for representing Clifford algebras as matrices, enabling automated calculations and proof verification.

Findings

Provides a systematic construction of matrix representations for Clifford algebras.

Develops an algorithm for computing multivector inverses using matrix representations.

Demonstrates the application of the Faddeev-LeVerrier-Souriau algorithm in this context.

Abstract

Modern advances in general-purpose computer algebra systems offer solutions to a variety of problems, which in the past required substantial time investments by trained mathematicians. An excellent example of such development are the Clifford algebras. The main objective of the paper is to demonstrate an utterly algorithmic construction of a Clifford algebra matrix algebra representation of a non-degenerate signature (p, q). While this is not the most economical way of implementation, it offers a transparent mechanism of translation between a Clifford algebra and its faithful real-valued matrix representation and can be used for automated proof checking. This representation is used to derive an algorithm for the computation of an arbitrary multivector inverse as a proof certificate. The proposed algorithm is a mapping of the Faddeev--LeVerrier--Souriau algorithm for computation of the characteristic polynomial of matrices.

Equations

$$\mathbf{B}:=\left\{P(E),\prec \right\}$$

$$A\ast B:={⟨AB⟩}_0$$

$$\mathbf{G}:=\left\{g_{\mu \nu }{e}_M\ast e_N|M\prec N\right\},g_{\mu }\in \{-1,0,1\}$$

$$A△B:=\left\{x:x\notin A\cap B,x\in A\cup B\right\}$$

$$\lambda +\mu e_1\dots e_k\in {\mathbb{G}}^n$$

$$e_k{e}_k={\sigma }_k{1}_{C\ell },{\sigma }_k\in \left\{1,-1,0\right\}$$

$$e_i{e}_j=-e_j{e}_i$$

$$1_{\mathbb{R}}=1_{C\ell }\equiv 1$$

$$P(E):=\{e_1,\dots ,e_n,e_1{e}_2,e_1{e}_3,\dots ,e_1{e}_2...e_n\}$$

$$V\left(E\right)\subset V\left(P(E)\right)$$

$$\lambda (e_a+e_b)=\lambda e_a+\lambda e_b$$

$$j:e_a{e}_b\dots e_m\mapsto e_J,J=\left\{a,b,\dots ,m\right\}$$

$$P:S_1\mapsto S_2$$

$$P:{\mathbf{B}}_1\mapsto {\mathbf{B}}_2$$

$$e_1\prec e_2\prec \underset{e_{12}}{\underbrace{e_1{e}_2}}\prec \dots \prec \underset{e_N}{\underbrace{e_1\dots e_n}}$$

$${\iota }_e:$$

$$\arg :f(g)\equiv f\circ g\mapsto g$$

$$M=s{P}_{\rho }\left\{e_{k_1}\dots e_{k_i}\right\}$$

$$e_{k_1}\dots e_{k_i}=$$

$$s_P\mathcal{R}\left(e_{k_{\alpha }}{e}_{k_{\beta }}\right)\mathcal{R}\left(e_{k_{\gamma }}\dots e_{k_{\omega }}\right)$$

$$\mathcal{R}\left(e_{k_1}\dots e_{k_i}\right)=s_P{e}_M,M=\left\{k_1△k_2△\dots k_{i-2p}\right\}$$

$$\mathcal{R}\left(e_{k_1}\dots e_{k_i}\right)=(-1{)}^q{s}_P{e}_M,M=\left\{k_1△k_2△\dots k_{i-2q}\right\}$$

$$\mathcal{R}\left(e_{k_1}\dots e_{k_i}\right)=0$$

$$\mu$$

$$\mathbf{M}:=\left\{m_{\mu \nu }\mathcal{R}(e_M{e}_N)|M\prec N\right\},m_{\mu \nu }=\left\{-1,0,1\right\}$$

$$\mathbf{M}=\left(\begin{array}[]{c|cccc}1&{e_{1}}&{e_{2}}&{e_{1}}{e_{2}}\\ \hline\cr{e_{1}}&1&{e_{1}}{e_{2}}&{e_{2}}\\ {e_{2}}&-{e_{1}}{e_{2}}&1&-{e_{1}}\\ {e_{1}}{e_{2}}&-{e_{2}}&{e_{1}}&-1\end{array}\right)$$

$$e_K{e}_J=m_{kj}{e}_s,S=K△J$$

$$K△J^{\prime }=K△J=S$$

Full text

The present work is funded in part by the European Union’s Horizon Europe program under grant agreement VIBraTE, grant agreement 101086815.

1. Introduction

Modern advances of general-purpose computer algebra systems offer solutions to a variety of problems, which in the past required substantial time investments by trained mathematicians. A good example for such development are the Clifford algebras. Recent years have seen renewed interest in Clifford algebra platforms. Some of these packages have been featured in peer-reviewed publications, notably, for Maple [1], Matlab [19], Mathematica [3] and Maxima [18]. There is also an inflow of new packages, such as Ganja.js for JavaScript [6], GaLua for Lua Galgebra for Python Grassmann for Julia.

The development of Clifford algebras is based on the insights of Grasssman, Hamilton and Clifford from the 19th century, yet at present these algebras are still an active area of mathematical research. Clifford algebras provide the natural generalizations of complex, dual and split complex (or hyperbolic) numbers into the concept of Clifford numbers. Since Clifford algebras are associative, they can be represented by matrices. Examples can be given by the Dirac’s $\gamma$ matrices, representing Space Time Algebra, and Pauli’s $\sigma$ matrices representing the three-dimensional (3D) algebra of the Euclidean space, that is the Geometric algebra of Clifford. In many applications, the Geometric algebra is referred to as the Algebra of Physical Space. The matrix representations have been studied before, resulting in somewhat complicated algorithmic constructions [4, 9, 13, 14, 15]. Recently, the question has been also addressed in [5, 11, 21].

The demonstrated application of the presented matrix representation algorithm is the computation of an inverse of a general multivector expression. This was achieved in two ways – either by going through an intermediate matrix representation and translating back the obtained result into a Clifford symbolical form or by mapping the Faddeev–LeVerrier–Souriau algorithm directly in the algebra itself. The paper was presented in preliminary form at the ICCA12 Conference, hosted virtually between 3–7 Aug 2020, at Hefei, China.

Results presented in the paper are derived using an elementary construction of Clifford algebra [12], which is suitable for direct implementation in a computer algebra system supporting symbolical transformations of expressions. From design perspective it was the preferred choice in the clifford package for the computer algebra system Maxima [16, 17, 18].

The paper is organized as follows. Sec. 2 introduces the notations and conventions. Sec. 3 presents a construction of the Clifford algebra, based on [12]. Sec. 4 introduces the sparse representation of the Clifford algebra. Sec. 5 studies the structure of the multiplication table of the algebra. Sec. 6 demonstrates the canonical matrix algebra representation theorem. Sec. 7 provides examples of low-dimensional matrix algebra representations. Sec. 9 demonstrates the Faddeev–LeVerrier–Souriau algorithm. The Appendices demonstrate supporting results.

2. Notation and Preliminaries

$C\ell_{n}$ will mean a Clifford algebra of order n but with unspecified signature. Algebra generators will be indexed by Latin letters. Multi-indices will be denoted with capital letters. The operation of taking k-grade part of an expression will be denoted by $\left\langle.\right\rangle_{k}$. The notation $(a;b)\ |\ \mathrm{pred}$ is interpreted as ”if the predicate pred is true then a else b”.

Matrices will be indicated with bold capital letters, while matrix entries will be indicated by lowercase letters.

Definition 1**.**

The extended basis set of the algebra will be defined as the ordered power set

[TABLE]

of all generators and their irreducible products.

Definition 2**.**

A multivector of the Clifford algebra is a linear combination of elements over the $2^{n}$-dimensional vector space spanned by $\mathbf{B}$.

Definition 3** (Scalar product).**

The scalar product of the blades $A$ and $B$ is defined as

[TABLE]

and extended by linearity to the entire algebra.

Definition 4** (Scalar product table).**

Define the scalar product table

[TABLE]

Definition 5** (Set difference).**

For the sets $A$ and $B$ define the symmetric difference as the operation

[TABLE]

3. Elementary construction of the algebra $C\ell_{p,q,r}$

Clifford algebras can be defined in several mathematically equivalent ways. However, from computer science perspective the preferred definition should be easy to implement with the tools of the computer algebra system of choice. This section gives a concise self-contained construction of the Clifford algebra based on Macdonald [12, 17]. The construction is repeated here for consistency of the presentation. The algebra is constructed by a set of generators, which can be identified with orthonormal vectors.

In order to construct the algebra we need some preliminary conventions. We assume that there are

• •

a fixed atomic generator symbol $\mathbf{e}$

• •

a set of $n$ abstract generators $E:=\left\{e_{1}\ldots e_{n}\right\}$ of the algebra produced by the action of the indicial map $\iota_{e}:k\mapsto e_{k}$ for $1\leq k\leq n$.

• •

a total order $\prec$ over $E$.

Definition 6** (Clifford algebra).**

The construction of the algebra proceeds in the following steps:

1. (1)

Define a vector space $V\left(E,\mathbb{R}\right)$ over the set of generators serving as basis with axioms

• •

Commutativity of vector addition: $a+b=b+a$.

• •

Associativity of vector addition: $(a+b)+c=a+(b+c)$

• •

Existence of additive unity [math]: $0+a=a+0=a$.

• •

For every vector, there exists an additive inverse: $a+(-a)=0$.

• •

Associativity of scalar multiplication: $\lambda(\mu a)=(\lambda\mu)a$

• •

Compatibility with scalar multiplication: $a\lambda=\lambda a$.

• •

Distributivity of scalar addition: $(\lambda+\mu)a=\lambda a+\mu a$.

• •

Distributivity of vector addition: $\lambda(a+b)=\lambda a+\lambda b$.

• •

Scalar multiplication identity: $1_{\mathbb{R}}\;a=a$.

for vectors $a,b,c$ and scalars $\lambda,\mu$. 2. (2)

Adjoin an associative algebra $\mathbb{G}(E,\mathbb{R})$ over $V\left(E,\mathbb{R}\right)$ using the Clifford product or geometric multiplication operation with axioms

• •

Existence of an algebra unity: $1_{C\ell}a=a1_{C\ell}=a$ for a non-scalar $a$.

• •

Left distributivity: $a(b+c)=ab+ac$ for arbitrary elements $a,b,c$.

• •

Right distributivity: $(a+b)c=ab+ac$ for arbitrary elements $a,b,c$.

• •

Associativity: $a(bc)=(ab)c$ for arbitrary elements $a,b,c$.

• •

Compatibility with scalars: $(\lambda a)(\mu b)=(\lambda\mu)(ab)$ for scalars $\lambda,\mu$ and non-scalar elements $a,b$. 3. (3)

Finally, assert

**Closure Axiom: **

For $k$ generators $e_{1},\ldots,e_{k}\in E$ and scalars $\lambda$, $\mu$ the multivector belongs to the algebra:

[TABLE]

**Reduction Axiom: **

For all generators

[TABLE]

where $\sigma_{k}$ are real or complex scalars. The signature set is $S:=\{\sigma_{k}\}$.

**Anti-Commutativity Axiom: **

For every two basis vectors, such that $e_{i}\prec e_{j}$

[TABLE]

We denote the algebra satisfying axioms 1-3 as Clifford algebra $C\ell_{p,q,r}$ , where $p$ elements of the orthonormal basis square to $1$, $q$ elements square to $-1$ and $r$ (degenerate) elements square to [math].

It should be remarked that the closure and compatibility Axioms are not included in the original construction of Macdonald. The algebra construction can be extended to other fields with characteristic zero. So-presented construction can be carried out without modifications in Computer Algebra systems like Maxima by defining proper simplification rules for the geometric product operation [16, 18].

The definition does not specify a particular ordering but merely asserts that an ordering must be used.

3.1. Consistency of the extension

Here we present results demonstrating the consistency of the Clifford algebra definition.

Proposition 1**.**

The scalar and algebra units coincide:

[TABLE]

Proof.

Suppose that $e_{0}\in V(E)$ is the unit of the algebra. Then by the anti-commutativity: $e_{0}e_{k}+e_{k}e_{0}=0\leftrightarrow e_{k}+e_{k}=0$ which is a contradiction. Therefore, $e_{0}\in\mathbb{K}\Rightarrow e_{0}=1$ by the uniqueness property. ∎

Theorem 1**.**

$V\left({E}\right)$* extends over the (ordered) power set of $E$.*

[TABLE]

That is $V\left(P(E)\right)$ is a vector space and we have the inclusion

[TABLE]

Proof.

The proof follows from the Closure Axiom.

Multiplicative properties:

1. (1)

Scalar commutativity $\mu e_{1}\ldots e_{k}=e_{1}\mu\ldots e_{k}=\ldots=e_{1}\ldots e_{k}\mu$ 2. (2)

In particular $1e_{1}\ldots e_{k}=e_{1}\ldots e_{k}1=e_{1}\ldots e_{k}$. 3. (3)

$0,e_{1}\ldots e_{k}=0\,e_{2}\ldots e_{k}=\ldots=0\,e_{k}=0$. 4. (4)

Scalar compatibility: $\lambda(\mu e_{1}\ldots e_{k})=\lambda(\mu e_{1})(e_{2}\ldots e_{k})=(\lambda\mu)e_{1}\ldots e_{k}$

And we extend by linearity and the closure over the whole algebra.

Additive properties:

1. (1)

Universality of zero: Let $L=0+e_{1}\ldots e_{k}$. Right-multiply by $e_{k}$. If $\sigma_{k}=0$ then the results follows trivially.

So let $\sigma_{k}\neq 0$. Then $Le_{k}=0e_{k}+e_{1}\ldots e_{k-1}\sigma_{k}=\sigma_{k}e_{1}\ldots e_{k-1}$. Right-multiply by $e_{k}$. Then $\sigma_{k}L=\sigma_{k}e_{1}\ldots e_{k}\Leftrightarrow L=e_{1}\ldots e_{k}$. 2. (2)

Scalar distributivity: $(c+d)e_{1}\ldots e_{k}=((c+d)e_{1})e_{2}\ldots e_{k}=(ce_{1}+de_{1})e_{2}\ldots e_{k}=ce_{1}\ldots e_{k}+de_{1}\ldots e_{k}$. 3. (3)

In particular, there is an additive inverse element $e_{1}\ldots e_{k}-e_{1}\ldots e_{k}=0$. 4. (4)

Commutativity of addition: Let $M=e_{a}+e_{b}-e_{b}-e_{a}$. Then for all possible orders of evaluation $M=0$. Therefore, by linearity $a+b=b+a$. 5. (5)

Associativity of addition: Let $M=(e_{a}+e_{b})+e_{c}-e_{a}-(e_{b}+e_{c})$. By commutativity of addition and closure $M=(e_{a}+e_{b})1-(e_{b}+e_{c})1-e_{c}-e_{a}$. Then for all possible orders of evaluation of the first two summands $M=e_{a}-e_{c}+e_{c}-e_{a}=0$. Therefore, by linearity $(a+b)+c=a+(b+c)$. 6. (6)

Distributivity of addition follows directly from the distributivity axioms and the closure axioms

[TABLE]

And we extend by linearity and the closure over the entire algebra. ∎

Based on this extension principle we can promote the indical map to a full index set isomorphism

[TABLE]

and use multi-index notation implicitly wherever appropriate. For consistency, we also extend the ordering to the power set $\prec_{P(E)}$, however no distinct symbol will be used for this.

Theorem 2** (Well-posedness).**

The Clifford algebra construction specified in Def. 6 is well-defined.

Proof.

$V(E)$ is well-defined. Moreover, the vector space extension is also well-defined. The compatibility Axioms imply the inclusion $V(E,\mathbb{R})\subset\mathbb{G}\left(E,\mathbb{R}\right)$.

Therefore, we only need to check the additional Algebra axioms. The Reduction Axiom agrees with the Closure Axiom trivially. The Anti-commutativity Axiom agrees with the Closure Axiom trivially. Further, by restriction to ordered pairs the Anti-commutativity Axiom does not apply when the Reduction Axiom applies. Therefore, these two axioms are logically independent. Therefore, restricted algebra construction over $V(E)$ is well-defined. The algebra has a maximal element by Prop. 6. The algebra extension is well-defined since the existence of $C\ell_{p+q+r=n}$ implies the existence of $C\ell_{p+q+r=n-1}$ by the Closure Axiom. By reduction, for $n=1$ the three possible cases for the sign of the $e_{1}^{2}$ represent the double, complex or dual numbers, respectively. Therefore, by induction $C\ell_{p+q+r=n}$ exists. ∎

Theorem 3** (Algebra equivalence).**

Two algebras with the same numbers of $p$, $q$ and $r$ generators are order-isomorphic.

Proof.

If the elements are ordered identically then the statement is trivial. Let’s assume that the elements are ordered differently. But then there is a permutation putting them into canonical order. Therefore, the algebras are isomorphic. Therefore, we can identify an order $\prec^{\prime}$ with the second permutation. ∎

Corollary 1** (S-Law of inertia).**

Denote by S the signature set of the algebra $C\ell$ . For two isomorphic algebras ${C\ell}^{\prime}$ and ${C\ell}^{\prime\prime}$ ( ${C\ell}^{\prime}\cong{C\ell}^{\prime\prime}$ ) there is an invertible map

[TABLE]

Conversely, if there is permutation

[TABLE]

then ${C\ell}^{\prime}\cong{C\ell}^{\prime\prime}$.

Proof.

The forward statement follows from Th. 3. Converse case: If the dimensions of $S_{1}$ and $S_{2}$ are equal and the numbers of $p$, $q$ and $r$ generators are equal then there is a permutation

[TABLE]

However, it can be noted that this is the same permutation which gives

[TABLE]

and hence by Th. 3 ${C\ell}^{\prime}\cong{C\ell}^{\prime\prime}$. ∎

Definition 7** (Canonical real algebra).**

Define the canonical ordering as the nested lexicographical order $\varrho$, such that $i<j\Longrightarrow e_{i}\prec e_{j}$ and extend it over $P(E)$ as:

[TABLE]

In addition assume that the first $p$ elements square to 1, the next $q$ elements square to -1 and the last $r$ elements square to 0. Then the algebra $C\ell_{p,q,r}$ specified by the structure $\left\{e,\varrho,\mathbb{R}\right\}$ is the canonical Clifford algebra.

Corollary 2**.**

The canonical algebra $C\ell_{p,q,r}$ defines an equivalence class with regard to permutations.

From computational perspective, last two corollaries are important because they indicate that an algorithm cane be implemented only for one Clifford algebra of a certain grade. They also imply that the order in which operations are composed can substantially simplify computations.

4. Indicial or sparse representation

Definition 8** (Indicial map).**

Define the indicial map $\iota_{e}$, acting on symbols by concatenation (i.e. of a set), such that

[TABLE]

where g is set-valued and assert the convention $\iota_{e}:\ \emptyset\mapsto 1_{Cl}$.

Definition 9**.**

Define the argument map $\mathrm{arg}$ acting on symbol compositions as

[TABLE]

Assert $\mathrm{arg}f=\emptyset$ for the atomic symbol $f$.

These definitions allow for stating a very general result about Clifford algebra representations.

Theorem 4** (Indicial representation).**

For generators $e_{s},e_{t}\in C\ell_{p,q,r}$, such that $s\neq t$, the following diagram commutes

{e_{s}}$${\{s\}}$${e_{s}e_{t}\equiv e_{st}}$${\{s,t\}}$$\mathrm{arg}$$\iota_{e}$$e_{t}$$\triangle\{t\}* *\mathrm{arg}$$\iota_{e}

Proof.

The right-left $\iota$ action follows from the construction of $C\ell_{p,q,r}$ . The left-right argument action is trivial. We observe that $\mathrm{arg}f=\emptyset$. Trivially, $\{s\}\triangle\emptyset=\emptyset\triangle\{s\}=\{s\}$. Let’s suppose that $s=t$. We notice that $\{s\}\triangle\{t\}=\{t\}\triangle\{s\}=\emptyset$. Let’s suppose that $s\neq t$. We notice that $\{s\}\triangle\{t\}=\{s,t\}$ and $\{t\}\triangle\{s\}=\{t,s\}$. ∎

This theorem can be used for the reduction of Clifford products.

4.1. Simplification of Clifford products

This section presents how to compute the canonical simplified form of blades. The non-simplified arbitrary Clifford product of multivectors will be referred to simply as a Clifford multinomial. The main lemma is demonstrated in [12] and is repeated here for convenience:

Lemma 1** (Permutation equivalence).**

Let $M=e_{k_{1}}\ldots e_{k_{i}}$ be a Clifford multinomial, where the indices are not necessarily different. Then

[TABLE]

where $s=\pm 1$ is the sign of permutation of $M$ and $P_{\rho}\left\{e_{k_{1}}\ldots e_{k_{i}}\right\}\mapsto e_{k_{\alpha}}\ldots e_{k_{\omega}}$ is the product permutation according to the canonical ordering.

Proof.

The proof follows directly from the anti-commutativity of Clifford multiplication for any two generator elements A-C, observing that the sign of a permutation of $S$ can be defined from its decomposition into the product of transpositions as $sgn(B)=(-1)^{m}$, where $m$ is the number of transpositions in the decomposition. ∎

Further, we can define a simplified form according to the action of the Reduction Axiom.

Theorem 5** (Reduced blade form).**

*Let $e_{k_{1}}\ldots e_{k_{i}}$ be an arbitrary Clifford multinomial, where the $i$ generators in $C\ell_{p,q,r}$ are not necessarily different. Let $s_{P}$ be the sign of the permutation $P_{\rho}\left\{e_{k_{1}}\ldots e_{k_{i}}\right\}$. We say that simplification induces a reduced or simplified form $\mathcal{R}$ with regard to permutations and product evaluations such that *

[TABLE]

Let $2p\leq i$ generators square as $e_{j}^{2}=1$. Then its simplified form is

[TABLE]

with all $k$-indices different. Let $2q\leq i$ generators square as $e_{j}^{2}=-1$. Then its simplified form is

[TABLE]

with all $k$-indices different. Let at lest $2$ generators square as $e_{j}^{2}=0$. Then its simplified form is

[TABLE]

Proof.

The product is transformed according to Lemma 1. Then we use Th 4 to compute the indices. By the Reduction axiom (R) the elements of equal indices are removed from the final index list. This process induces a factor of $(-1)^{q}$ in the final result. Hence, for $2q$ elements of index $a$ $\left\{k_{a}\right\}\triangle\left\{k_{a}\right\}=\emptyset$. Further $\emptyset\triangle\left\{k_{i}\right\}=\left\{k_{i}\right\}\triangle\emptyset=\left\{k_{i}\right\}$. The final list can be translated back into a blade form by the map $\iota_{e}$ by Th. 4. The associativity of $\mathcal{R}$ follows from the associativity of the Clifford product. ∎

The form equivalence is used for simplification of expressions in the Clifford Maxima package.

5. Multiplication tables of Clifford algebras

Definition 10** (Full Matrix multiplication table).**

Consider the extended basis $\mathbf{B}$. Define the multiplication table matrix as the mapping of the basis into the set of square matrices valued in $C\ell_{p+q+r}$ :

[TABLE]

with matrix consisting of the ordered ans simplified (i.e. $\mathcal{R}$ form) product entries using the multi-index notation

[TABLE]

Since $\mathbf{M}$ is a square matrix we will not make a distinction for its dimensions, so $dim(\mathbf{M})$ will mean $2^{n}\times 2^{n}$ or $2^{n}$ depending on the context.

Example 1**.**

Consider $C\ell_{2,0,0}$. The full matrix multiplication table is

[TABLE]

The lines are drawn for clarity.

Proposition 2**.**

Consider the multiplication table $\mathbf{M}$. All elements $m_{kj}$ are different for a fixed row $k$. All elements $m_{iq}$ are different for a fixed column $q$.

Proof.

The results follow from the simplification Th. 5. Fix $k$. Then for $e_{K},e_{J}\in\mathbf{B}$

[TABLE]

which is a different way of writing the simplification Lemma. Suppose that we have equality for 2 indices $j,j^{\prime}$. Then

[TABLE]

and let $\delta=J\cap J^{\prime}$. Then

[TABLE]

Therefore, $j=j^{\prime}$. By symmetry, the same reasoning applies to a fixed column $q$. ∎

Lemma 2** (Multiplication Matrix Structure).**

For the multi-indices $S\prec T$, such that $S\cap T=\emptyset$, the following implications for the elements of $\mathbf{M}$ hold:

{m_{\mu\lambda}\,e_{s}}$${m_{\mu\lambda^{\prime}}e_{t}}$${m_{\lambda\mu}e_{s}}$${m_{\lambda\mu}m_{\mu\lambda^{\prime}}e_{st}}$${m_{\lambda\lambda^{\prime\prime}}e_{st}}$$\exists$$\exists\lambda^{\prime}>\lambda$$\exists$$\exists\lambda^{\prime\prime}=\lambda^{\prime}

so that

[TABLE]

for some index $\mu$.

Proof.

By the properties of $\mathbf{M}$ there exists $\lambda^{\prime}>\lambda$, such that $e_{M}e_{L^{\prime}}=m_{\mu\lambda^{\prime}}\,e_{t},\ \ L^{\prime}\backslash M=T$ for $L\prec L^{\prime}$. Choose $M$, s.d. $L\prec M\prec L^{\prime}$. Then for $L\prec M\prec L^{\prime}$ and $S\prec T$

[TABLE]

Suppose that $e_{s}e_{t}=e_{st},st=S\cup T=S\triangle T$. Multiply together the diagonal nodes in the matrix

[TABLE]

Therefore, $s\in L$ and $t\in L^{\prime}$. We observe that there is at least one element (the algebra unity) with the desired property $\sigma_{\mu}\neq 0$.

Further, we observe that there exists unique $\lambda^{\prime\prime}$ such that $m_{\lambda\lambda^{\prime\prime}}e_{st}$. Since $\lambda$ is fixed by Th. 5 this implies that $L^{\prime\prime}=L^{\prime}\Rightarrow\lambda^{\prime\prime}=\lambda^{\prime}$. Therefore,

[TABLE]

which implies the identity

[TABLE]

∎

6. Clifford algebra real matrix representation theorem

In the present article we will focus on non-degenerate Clifford algebras. Therefore, we assume that the product set

[TABLE]

is valued in the set $\left\{-1,1\right\}$ if not stated otherwise.

In order to state the main result we need the following definitions:

Definition 11** (Coefficient map).**

Define the linear map acting element-wise

[TABLE]

by the action

[TABLE]

Define the coefficient map indexed by the multi-index $S$ as

[TABLE]

Definition 12** (Canonical matrix map).**

For the multi-index $S$ define the map

[TABLE]

where s is the ordinal of $e_{S}$ in the multivector basis $\mathbf{B}$ and $\mathbf{A}_{S}$ is computed as in Def. 11. Further, denote the set of all maps as $\pi=\left\{\pi_{s}\right\}$ and let $\pi_{s}\equiv\pi(e_{s})$.

Proposition 3**.**

The $\pi$-map is linear.

The proof follows from the linearity of the coefficient map and matrix multiplication.

Theorem 6** (Semigroup property).**

Let $e_{s}$ and $e_{t}$ are basis elements. Then the map $\pi$ acts on $C\ell_{p,q}$ according to the following diagram

{e_{s}}$${\mathbf{E}_{s}}$${e_{s}e_{t}\equiv e_{st}}$${\mathbf{E}_{st}\equiv\mathbf{E}_{s}\mathbf{E}_{t}}$$e_{t}* \pi$$\mathbf{E}_{t} *$\pi$

The map $\pi$ distributes over the Clifford product:

[TABLE]

The set of all matrices $\mathbf{E}_{s}$ forms a semigroup.

Proof.

Let

[TABLE]

We specialize the result of Lemma 2 for $S=\{s\}$ and $T=\{t\}$ and observe that

[TABLE]

for $\lambda,\lambda^{\prime},\mu\leq n$

[TABLE]

Therefore,

[TABLE]

Moreover, we observe that

[TABLE]

For the semi-group property, consider that since $\pi$ is linear it is invertible. Since $\pi$ distributes over Clifford product its inverse $\pi^{-1}$ distributes over matrix multiplication:

[TABLE]

However, $C\ell_{p,q}$ is closed by construction, therefore, the set $\left\{\mathbf{E}\right\}_{s}$ is closed under matrix multiplication. ∎

Corollary 3**.**

For $C\ell_{p,q}$ the set of all matrices $\mathbf{E}_{s}$ forms a group, which is a representation of the Pin group $\mathrm{Pin}_{p,q}$.

Proof.

We observe that for non degenerate algebras every generator has an inverse element by Props. 8; therefore for non-degenerate algebras $C\ell_{p,q}$

[TABLE]

However, $\sigma$ is valued in $\{-1,+1\}$. ∎

Proposition 4**.**

Let $\mathbf{L}:=\left\{l_{i}\right\},l_{i}\in P(E)$ and $\mathbf{R}_{s}$ is the first row of $\mathbf{E}_{s}$. Then $\pi^{-1}:\mathbf{E}_{s}\mapsto\mathbf{R}_{s}\mathbf{L}$.

Proof.

We observe that by the Prop. 9 the only non-zero element in the first row of $\mathbf{E}_{s}$ is $\sigma_{1}m_{1s}=1$. Therefore, $\mathbf{R}_{s}\mathbf{L}=e_{s}$. ∎

The main result of the section is stated in the theorem below.

Theorem 7** (Canonical Real Matrix Representation).**

Define the map $g:\mathbf{A}\mapsto\mathbf{G}\mathbf{A}$. Then

[TABLE]

so that the diagram commutes

{\mathbf{M}}$${\mathbf{A}_{s}}$${\mathbf{G}\mathbf{M}}$${\mathbf{E}_{s}=\mathbf{G}\mathbf{A}_{s}}$$g* C_{s}$$g *C_{s}$$\pi_{s}

Further, $\pi$ is an isomorphism inducing a Clifford algebra representation in the real matrix algebra:

{C\ell_{p,q}(\mathbb{R})}$${C\ell_{p,q}\left[\mathbf{Mat}_{\mathbb{R}}({2^{n}}\times{2^{n}})\right]}$$\pi$$\pi^{-1}

The $\pi$ map distributes over the Clifford product (homomorphism):

[TABLE]

Proof.

The $\pi$-map is a linear isomorphism. The set $\left\{\mathbf{E}_{s}\right\}$ forms a group, which is a subset of the matrix algebra $\mathbf{Mat}_{\mathbb{R}}(N\times N),N=2^{n}$. Let $\pi(e_{s})=\mathbf{E}_{s}$ and $\pi(e_{t})=\mathbf{E}_{t}$. It is claimed that

1. (1)

$\mathbf{E}_{s}\mathbf{E}_{t}\neq\mathbf{0}$ by the Sparsity Lemma 3. 2. (2)

$\mathbf{E}_{s}\mathbf{E}_{t}=-\mathbf{E}_{t}\mathbf{E}_{s}$ by Prop. 7. 3. (3)

$\mathbf{E}_{s}\mathbf{E}_{s}=\sigma_{s}\mathbf{I}$ by Prop. 8.

Therefore, the set $\left\{\mathbf{E}_{s}\right\}_{s=1}^{N}$ is an image of $C\ell_{p,q}$ . ∎

So-constructed canonical matrix representation has an interesting structure. This can be summarized in the following results.

Proposition 5**.**

Consider the non-degenerate algebra $C\ell_{p,q}$ . Then for any element $e_{s}$ :

• •

If $\sigma_{s}=1$ then $\mathbf{E}_{s}$ is symmetric.

• •

If $\sigma_{s}=-1$ then $\mathbf{E}_{s}$ is anti-symmetric.

Proof.

[TABLE]

Then for $\sigma_{s}=1\Rightarrow\mathbf{E}_{s}-\mathbf{E}_{s}^{T}=\mathbf{0}$, therefore $\mathbf{E}_{s}$ is symmetric. While for $\sigma_{s}=-1\Rightarrow-\mathbf{E}_{s}-\mathbf{E}_{s}^{T}=-(\mathbf{E}_{s}+\mathbf{E}_{s}^{T})=\mathbf{0}$, therefore $\mathbf{E}_{s}$ is anti-symmetric. ∎

As can be seen from the next examples the structure of the matrices is very symmetric. The diagonal holds the scalar value

[TABLE]

while the co-diagonal holds the pseudoscalar value,

7. Low-dimensional matrix representations

In this section examples are given for non-degenerate Clifford algebras.

7.1. Representations for n=2

For a general element of the form

[TABLE]

we have the following canonical matrix representations:

• •

Euclidean algebra $C\ell_{2,0}$

[TABLE]

• •

Split-quaternions $C\ell_{1,1}$

[TABLE]

• •

Quaternions $C\ell_{0,2}$

[TABLE]

7.2. Representations for n=3

For a general element of the form

[TABLE]

we have the following canonical matrix representations:

• •

Geometric algebra / Algebra of Physical Space $C\ell_{3,0}$

[TABLE]

• •

Clifford algebra $C\ell_{2,1}$

[TABLE]

• •

Clifford algebra $C\ell_{1,2}$

[TABLE]

• •

Clifford algebra $C\ell_{0,3}$

[TABLE]

7.3. Representations for n=4

For a general element of the form

[TABLE]

we have the following canonical matrix representations:

• •

Euclidean algebra $C\ell_{4,0}$

[TABLE]

• •

Clifford algebra $C\ell_{3,1}$

[TABLE]

• •

Clifford algebra $C\ell_{2,2}$

[TABLE]

• •

Space-time algebra $C\ell_{1,3}$

[TABLE]

• •

Clifford algebra $C\ell_{0,4}$

[TABLE]

Computations have been carried out using the computer code included in the Appendix.

8. Complex matrix representation of central algebras

In the present section we exhibit an algorithm for the construction of the complex matrix representation of central Clifford algebras. As an example, consider the dual representation of the $C\ell_{3,0}$ , where $I^{2}=-1$. The matrix multiplication table can be represented as

[TABLE]

where we have substituted the pseudoscalar $I$ for the imaginary scalar unit $i$. The main diagonal blocks do not contain i, while the co-diagoanl ones do. $\mathbf{M}$ can be further transformed into

[TABLE]

where this structure is not obvious, but we are left only with the generators. Therefore, the same representation algorithm can be applied. We use the complexified set of generators

[TABLE]

and compute the reduced multiplication tables $-\mathbf{M}_{22}=\mathbf{Mat}(iE)$ and $\mathbf{G}=diag\left\langle-\mathbf{M}_{22}\right\rangle$. The general algebra element can be represented as

[TABLE]

Therefore, by linearity $\pi(A)=\pi(a+v)+\pi(I)\pi(b+p)$. Therefore, we can make a substitution $\pi(I)\mapsto i$ and proceed as before. In such way, the complex matrix representation will be given by

[TABLE]

Therefore, one can arrive at the following representation:

[TABLE]

The advantage of this representation is that the trace produces the pseudoscalar plus the scalar components in an obvious manner $\mathrm{Tr}\mathbf{A}=4\left({a_{1}}-i{a_{8}}\right)$.

9. Computation of multivector inverses

Computation of Clifford inverse has drawn attention in the literature [2, 10, 20]. Multivector inverses can be computed using the matrix representation and the characteristic polynomial. The matrix inverse is

[TABLE]

where $\#$ denotes the matrix adjunct operation and $||\mathbf{A}||$ denotes the matrix determinant. The formula is not practical, because it requires the computation of $n^{2}+1$ determinants. With the help of the Cayley-Hamilton Theorem, the inverse of A can be expressed as a polynomial in A. The inverse can be computed as the last step of the Faddeev–LeVerrier–Souriau (FVS) algorithm [22, 7]. The algorithm computes the coefficients of the characteristic polynomial

[TABLE]

in n steps:

[TABLE]

with $\mathbf{M}_{n}=\mathbf{0}$. The matrix inverse can be computed from the last step of the algorithm as

[TABLE]

under the obvious restriction $t_{n}\neq 0$.

Example 2**.**

In $C\ell_{2,0}$ the general multivector inverse representation is

[TABLE]

The corresponding characteristic polynomial is

[TABLE]

Example 3**.**

In the split quaternions $C\ell_{1,1}$ the general inverse representation is

[TABLE]

The corresponding characteristic polynomial is

[TABLE]

Example 4**.**

In the quaternions the general multivector inverse representation is

[TABLE]

corresponding to the characteristic polynomial

[TABLE]

The FVS algorithm has a direct representation in terms of Clifford multiplications as demonstrated by the next result.

Theorem 8** (Multivector inversion algorithm).**

Suppose that $A\in C\ell_{p,q}$ is a multivector of maximal grade $r\leq p+q$. The Clifford inverse, if it exists, can be computed by the algorithm in $k=2^{r}$ steps as

[TABLE]

until the step where $m_{k}=0$ so that

[TABLE]

The inverse does not exist if $t_{k}=0$. The characteristic polynomial is

[TABLE]

Proof.

The proof follows from the distributivity of the $\pi$ map over the matrix and Clifford products. Furthermore, $Tr[\mathbf{M}_{k}]=N\left\langle m_{k}\right\rangle_{0}$. Moreover, the FVS algorithm terminates with $\mathbf{M}_{N}=0$, which corresponds to $N=2^{p+q}$.

Suppose that $A$ is of maximal grade $r$. Let $E$ be the set of all blades of grade less or equal to $r$. We compute the restricted multiplication tables $\mathbf{M}(E_{r})$ and respectively $\mathbf{G}(E_{r})$ and form the restricted map $\pi_{r}$. Then

[TABLE]

Therefore, the FVS algorithm will terminate in $k=2^{r}$ steps. Now, suppose that $t_{k}\neq 0$, then

[TABLE]

∎

9.1. Low-dimensional inverses

While inverses can be computed for an arbitrary dimensions the expressions quickly become extremely cumbersome. We will consider $C\ell_{3,0}$ . Up to sign permutations the results are expected to hold also for $C\ell_{2,1}$ , $C\ell_{1,2}$ , and $C\ell_{0,3}$ . Let

[TABLE]

Then the application of the LVS algorithm yields

[TABLE]

where the determinant is given by

[TABLE]

and

[TABLE]

While the vector part is given by

[TABLE]

the bi-vector part by

[TABLE]

and the pseudoscalar part by

[TABLE]

The inverse exists if the determinant $\Delta\neq 0$. The characteristic polynomial is

[TABLE]

9.2. Numerical examples

Example 5**.**

Let us compute an example in $C\ell_{3,1}$ , having rational coefficients. Let

[TABLE]

The computed inverse is

[TABLE]

The intermediate steps of the computations are available for inspection by setting the appropriate flags in the code. However, the rational numbers involved quickly become very long.

Let us compute several examples in $C\ell_{5,2}$ .

Example 6**.**

Let

[TABLE]

Then

[TABLE]

Example 7**.**

Let

[TABLE]

Then

[TABLE]

Let

[TABLE]

Then

[TABLE]

Example 8**.**

In $C\ell_{2,2}$ let

[TABLE]

[TABLE]

so that

[TABLE]

On the other hand, for certain low-dimensional cases – n=2 and n=4, the FLV algorithm can be simplified as follows.

[TABLE]

leading to the same result for $A^{-1}$.

10. Discussion

The main theoretical objective of the present paper was to demonstrate a universal construction of a matrix algebra representation of Clifford algebra of non-degenerate arbitrary signature (p, q). This is achieved by exhibiting an explicit isomorphism between a given Clifford algebra and its faithful matrix representation. Instrumental for the presented approach is the Clifford algebra matrix multiplication table $\mathbf{M}$. The structure of this table is constrained by the structure of the Clifford algebra itself, which allows for deriving useful relations between the matrix entries thus reducing the computational complexity of the matrix construction. As demonstrated further in the paper, the matrix multiplication table $\mathbf{M}$ encodes all properties of the algebra It should be noted that such a representation is not optimal from data compression perspective. On the other hand, it can be used for automatic computer code generation for the lower-dimensional algebras. Target applications of such an approach can be computational environments, such as Matlab, or computer code preprocessors, such as Gaalop for C++ [8].

It should be noted that the FLV algorithm can result easily in float precision issues. On the other hand, the FLV algorithm is in fact a proof certificate, that is it terminates in a finite number of steps and if an inverse exits it provides it.

Funding

Funding for the initial clifford development was provided by a grant from Research Fund – Flanders (FWO), contract numbers G.0C75.13N, VS.097.16N.

Conflicts of interest

The author declares no competing intesrests.

Availability of data and material

Data are available in a Zenodo repository [16].

Code availability

Maxima code is available in a Zenodo repository [16].

Appendix A Auxiliary results

Proposition 6** (Maximal element).**

The algebra $C\ell_{p,q,r}$ has a maximal element $\mathcal{I}=e_{1}\ldots e_{n}$ called pseudoscalar.

Proof.

Suppose that $r>0$. Then if a product contains more than one nilpotent generators of the same index the simplified form is 0 by Lemma 1. If a product contains exactly one nilpotent generator per index the maximal element is the product of all generators by Th. 5.

[TABLE]

Suppose that $r=0$. Then by Th. 5 the maximal element is the product of all generators

[TABLE]

This element is referred to as the pseudoscalar of the algebra. ∎

Proposition 7**.**

For generator elements $e_{s}$ and $e_{t}$

[TABLE]

Proof.

Consider the basis elements $e_{s}$ and $e_{t}$. By linearity and Clifford-product distributivity of the $\pi$ map

[TABLE]

Therefore, for vector elements

[TABLE]

∎

Proposition 8**.**

$\mathbf{E_{s}}\mathbf{E_{s}}=\sigma_{s}\mathbf{I}$* *

Proof.

Consider the matrix $\mathbf{W}=\mathbf{G}\mathbf{A_{s}}\mathbf{G}\mathbf{A_{s}}$. Then element-wise $w_{\mu\nu}=\sum_{\lambda}\sigma_{\mu}\sigma_{\lambda}a_{\mu\lambda}a_{\lambda\nu}$. By Lemma 3 $\mathbf{W}$ is sparse so that $w_{\mu\nu}=(0;\sigma_{\mu}\sigma_{q}a_{\mu q}a_{q\nu})$.

From the structure of $\mathbf{M}$ for the entries containing the element $e_{S}$ we have the equivalence

[TABLE]

After multiplication of the equations we get

[TABLE]

which simplifies to the First fundamental identity:

[TABLE]

We observe that if $\sigma_{\mu}=0$ or $\sigma_{q}=0$ the result follows trivially. In this case also $\sigma_{s}=0$.

Therefore, let’s suppose that $\sigma_{s}\sigma_{q}\sigma_{\mu}\neq 0$. We multiply both sides by $\sigma_{s}\sigma_{q}\sigma_{\mu}$

[TABLE]

However, the RHS is a diagonal element of $\mathbf{W}$, therefore by the sparsity it is the only non-zero element for a given row/column so that

[TABLE]

∎

Appendix B Properties of the scalar product multiplication tables

Theorem 9**.**

The scalar product table is a diagonal matrix $\mathbf{G}$, which is invariant under orthogonal transformations.

Proof.

The proof is based on Macdonald [12]. From the definition of the scalar product it is obvious that $\mathbf{G}$ is diagonal. Consider the orthogonal transformation $\mathbf{F}=\mathbf{A}\mathbf{E}$ with orthogonal matrix $\mathbf{A}$. We evaluate $\mathbf{F}^{T}\mathbf{F}$. Then element-wise (summation by repeated indices)

[TABLE]

Then for $m=n$

[TABLE]

by the orthogonality of entries. Then $diag\left\langle\mathbf{F}^{T}\mathbf{F}\right\rangle=\mathbf{G}$, where diag denotes the diagonal elements. ∎

Definition 13** (Sparsity property).**

A matrix has the sparsity property if it has exactly one non-zero element per column and exactly one non-zero element per row. Such a matrix we call sparse.

Proposition 9**.**

For $e_{s}\in\mathbf{E}$ the matrix $\mathbf{A}_{s}=C_{e_{s}}(\mathbf{M})$ is sparse.

Proof.

Fix an element $e_{s}\in\mathbf{E}$. Consider a row $k$. By prop 2 there is a $j$, such $e_{kj}=e_{s}$. Then $a_{kj}=\mu_{kj}$, while for $i\neq j$ $a_{ki}=0$. Consider a column $m$ By prop 2 there is a $j$, such $e_{jm}=e_{s}$. Then $a_{jm}=\mu_{jm}$, while for $i\neq j$ $a_{im}=0$. Therefore, $\mathbf{A}_{s}$ has the sparsity property. ∎

Lemma 3** (Sparsity lemma).**

If the matrices $\mathbf{A}$ and $\mathbf{B}$ are sparse then so is $\mathbf{C}=\mathbf{AB}$. Moreover,

[TABLE]

(no summation!) for some index $q$.

Proof.

Consider two sparse square matrices $\mathbf{A}$ and $\mathbf{B}$ of dimension $n$. Let $c_{ij}=\sum_{\mu}a_{i\mu}b_{\mu j}$. Then as we vary the row index $i$ then there is only one index $q\leq n$, such that $a_{iq}\neq 0$. As we vary the column index $j$ then there is only one index $q\leq n$, such that $b_{qj}\neq 0$. Therefore, $c_{ij}=(0;a_{iq}b_{qj})$ for some $q$ by the sparsity of $\mathbf{A}$ and $\mathbf{B}$.

As we vary the row index $i$ then $c_{qj}=0$ for $i\neq q$ for the column $j$ by the sparsity of $\mathbf{A}$. As we vary the column index $j$ then $c_{iq}=0$ for $j\neq q$ for the row $i$ by the sparsity of $\mathbf{B}$. Therefore, $\mathbf{AB}$ is sparse. ∎

Corollary 4**.**

If $\mathbf{A}$ is sparse then $\mathbf{A^{2}}$ is diagonal.

Proof.

Let $\mathbf{C}=\mathbf{A^{2}}$ We observe that by sparsity $a_{iq}b_{qj}\neq 0$ if $i=j$. ∎

B.1. Non-degenerate algebras

Proposition 10**.**

For non degenerate algebras $C\ell_{p,q}$ the matrix $\mathbf{G}$ is orthogonal and

[TABLE]

Proof.

$\mathbf{G}=\mathbf{G}^{T}$ by symmetry. Since its elements are valued $\pm 1$ then element-wise $\sum_{\lambda}g_{\mu\lambda}g_{\lambda\mu}=\sigma_{\mu}^{2}=1$ ∎

Proposition 11**.**

For non-degenerate algebras $C\ell_{p,q}$ $\mathbf{E_{s}^{-1}}=\sigma_{s}\mathbf{E_{s}}$.

Theorem 10**.**

For non-degenerate algebras $C\ell_{p,q}$ the matrices $\mathbf{A}_{s}$ and $\mathbf{E}_{s}$ are orthogonal.

Proof.

First we prove an auxiliary equation. Consider the matrix

[TABLE]

Element-wise $y_{\mu\nu}=\sigma_{\mu}a^{s}_{\mu\nu}-\sigma_{s}a^{s}_{\nu\mu}\sigma_{\nu}$. Suppose that $a^{s}_{\nu\mu}\neq 0$. Then

[TABLE]

Therefore,

[TABLE]

Right multiply by $\mathbf{GA}_{s}$:

[TABLE]

so that

[TABLE]

Since $\mathbf{G}$ is orthogonal the second assertion follows as well. ∎

Corollary 5** (Second fundamental identity).**

Consider the element $e_{s}$ in a non-degenerate algebra $C\ell_{p,q}$ . Then there is an index $q\leq n=p+q$, such that

[TABLE]

Proof.

$\mathbf{A}_{s}{\mathbf{A}_{s}^{T}}=\mathbf{I}\Rightarrow(a^{s}_{q\mu})^{2}=1$ for some $q$. $\mathbf{E}_{s}{\mathbf{E}_{s}^{T}}=\mathbf{I}\Rightarrow\sigma_{\mu}\sigma_{q}(a^{s}_{q\mu})^{2}=\sigma_{\mu}\sigma_{q}=1$ for some $q$. The final assertion follows from the first fundamental identity. ∎

B.2. Degenerate algebras

For degenerate algebras a weaker version of the result holds

Proposition 12**.**

Consider a degenerate algebra $C\ell_{p,q,r}$ and let $\mathbf{H}:=\mathbf{G}^{2}$. Then $\mathbf{H}$ is idempotent.

Proof.

We observe that since $\mathbf{G}$ is diagonal and $\sigma_{i}^{2}=1$ it follows that $\mathbf{H}$ contains only 1’s and 0’s on its diagonal so $h_{\mu\lambda}h_{\lambda\mu}=h_{\mu\mu}^{2}\in\{0,1\}$ and

[TABLE]

∎

Appendix C Maxima code

The code was implemented using the Maxima package clifford developed by the author [16, 18]. Examples presented above can be recomputed using the command elem2mat1 after proper initialization of the package. The key functions of the implementation are listed below.