Geometric Matrices of Dual Null Vectors

Abstract

A (Clifford) geometric algebra is usually defined in terms of a quadratic
form. A null vector v is an algebraic quantity with the property that v 2 =
0. Null vectors over the real or complex numbers are taken as fundamental
and are added and multiplied together according to the familiar rules of
real or complex square matrices. In a series of ten definitions, the concepts
of a Grassmann algebra, its dual Grassmann algebra, the associated real
and complex geometric algebras, and their isomorphic real or complex
coordinate matrix algebras are laid down.

Full text

Geometric Matrices of Dual Null Vectors
Garret Sobczyk
Universidad de las Am´ericas-Puebla
Departamento de Actuara F´ısica y Matem´aticas
72820 Puebla, Pue., M´exico
https://www.garretstar.com/
May 19, 2020
Abstract
A (Clifford) geometric algebra is usually defined in terms of a quadratic
form. A null vector vis an algebraic quantity with the property that v2=
0. Null vectors over the real or complex numbers are taken as fundamental
and are added and multiplied together according to the familiar rules of
real or complex square matrices. In a series of ten definitions, the concepts
of a Grassmann algebra, its dual Grassmann algebra, the associated real
and complex geometric algebras, and their isomorphic real or complex
coordinate matrix algebras are laid down.
AMS Subject Classification: 15A63, 15A66, 15A75, 46E40
Keywords: Clifford algebra, geometric algebra, Grassmann algebra, ma-
trix algebra, null vector.
1 Introduction
The development of the concept of duality in mathematics has a robust history
dating back more than 100 years, and involving 20th Century mathematicians
of the first rank such as F. Reisz and S. Banach, but also encompassing first
rank 19th Century mathematicians such as Gauss, Lobachevsky and Bolyai. A
fascinating history of the seminal Hahn-Banach Theorem, and all its ramifica-
tions regarding the issue of duality in finite and infinite dimensional Hilbert
spaces is given in [3].
Considering infinite dimensional vector spaces broadens and immensely deep-
ens the mathematical issues involved in the concept of duality [1, 5]. In this
paper we consider duality only in regard to a finite dimensional vector space,
where it is well known that duality is equivalent to defining a Euclidean inner
product. The purpose of this paper is to show how defining an inner product
on a vector space of null vectors, over the real or complex numbers, not only
captures the notion of duality but also nails down the corresponding isomorphic
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real or complex matrix algebra of a geometric algebra. In a series of 10 defi-
nitions, we define a Grassmann algebra, its dual Grassmann algebra and their
associated Clifford geometric algebra. The isomorphic real or complex coor-
dinate matrices of a geometric algebra, called geometric matrices, are directly
constructed alongside and serve as a powerful computational and pedagogical
tool.
2 Ten Definitions
1. Null vectors are algebraic quantities x6= 0 with the property that x2=
0. They are added and multiplied together using the same rules as the
addition and multiplication of real or complex square matrices. The trivial
null vector is denoted by 0.
2. Two null vectors a1and a2are said to be anticommutative if
a1a2+a2a1= 0.
3. A set of mutually anticommuting null vectors {a1, . . . , an}Fis said to be
linearly independent over F=Ror C, if a1···an6= 0.1In this case they
generate the 2n-dimensional Grassmann algebra
Gn(F) := gen{a1, . . . , an}F.
4. Let Ai:= 1
aifor i= 1, . . . , n. By the right directed Kronecker prod-
uct, G2(F) := A1−→
⊗A2:= 


1
a1
a2
a12


. More generally, the right directed
Kronecker product
Gn(F) := A1−→
⊗ ···−→
⊗An
gives a 2n-column matrix of the basis elements
{1; a1, . . . , an;. . . ;aλ1···aλk;. . . ;a1···an}F
of the 2n-dimensional Grassmann algebra Gn(F). The n
kelements
aλ1···λk:= aλ1···aλk
for 1 ≤λ1<· ·· < λk≤nare called k-vectors. Similarly, the left directed
Kronecker product
G2(F) := AT
2←−
⊗AT
1:= (A1−→
⊗A2)T= ( 1 a1a2a21 ),
1More general fields Fcan be considered as long as characteristic F 6= 2. For an interesting
discussion of this issue see [2].
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and more generally,
Gn(F) := (AT
n←−
⊗ ···←−
⊗AT
1) := (A1−→
⊗ ···−→
⊗An)T,
gives a 2n-row matrix of the 2nbasis of the Grassmann algebra Gn(R),
[2,p. 82].
5. A pair of null vectors aand bare said to be algebraically dual if
ab +ba = 1.
In this case, we define a∗:= band b∗:= a, from which it follows that
(a∗)∗=b∗=a.
6. Let Gn(R) := A1−→
⊗ ··· −→
⊗Anand G#
n(R) = BT
n←−
⊗ ··· ←−
⊗BT
1be two 2n-
dimensional real Grassmann algebras. The Grassmann algebras Gn(R)
and G#
n(R) are said to be dual Grassmann algebras if there exists gener-
ating bases
Gn(R) := gen{a1, . . . , an}Rand G#
n(R) := gen{b1, . . . , bn}R
such that
2ai·bj:= aibj+bjai=δij .
In this case, G∗
n(R) := G#
n(R). Of course, nothing is surprising because it
is well known that any standard vector space V, and it dual space V∗can
be represented in terms of an equivalent inner product.
7. The real geometric algebra
Gn,n(R) := Gn(R)⊗G∗
n(R) = gen{a1, . . . , an, b1, . . . , bn}
where for i, j = 1, . . . , n,
2ai·bj=aibj+bjai=δij .
8. Defining the idempotents ui:= biai, the quantity
u1···n:= u1··· un=
n
Y
i=1
biai=b1a1··· bnan
is a primitive idempotent in the geometric algebra Gn,n. The nilpotent
matrix basis of the geometric algebra Gn,n is specified by
Gn,n := A1−→
⊗ ··· −→
⊗Anu1···nBT
n←−
⊗ ··· ←−
⊗BT
1=−→
⊗n
i=1Aiu1···n−→
⊗n
i=1Ai∗
,
where A∗
i:= BT
ifor i= 1, . . . , n. In the nilpotent matrix basis, any
g∈Gn,n is explicitly expressed in terms of its coordinate geometric matrix
[g] := [gij ] for gij ∈R, by
g=−→
⊗n
i=1AT
iu1···n[g]←−
⊗1
i=nBi,
[7, Chapter 5].
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9 . The standard basis of Gn(R) = Gn,n(R), is specified by
Gn,n := R(e1, . . . , en, f1, . . . , fn) = gen{e1, . . . , en, f1, . . . , fn}R
where ei:= ai+biand fi:= ai−bifor i= 1, . . . , n. The basis vectors are
mutually anticommutative and satisfy the basic property
e2
i= 1,and f2
i=−1
for i= 1, . . . , n, as can be easily verified.
10. The real geometric algebra
Gn,n+1 := R(e1, . . . , en, f1, . . . , fn+1) = Gn(C)
where the imaginary unit
i=√−1 := (e1f1)···(enfn)fn+1 ⇐⇒ fn+1 := i(e1f1)···(enfn).
How geometric matrices arise as algebraically isomorphic coordinate matri-
ces, for the simplest case of 2 ×2 real or complex matrices, and their practical
application to the classical Pl¨ucker relations is given in [6]. A general intro-
duction to geometric algebras and their coordinate geometric matrices is given
in [7]. A periodic table of all of the classical Clifford geometric algebras is de-
rived from three Fundamental Structure Theorems in [8]. Other links to relevant
research can be found on my website <https : //www.garretstar.com/ >.
Acknowledgements
The seeds of this paper were planted almost 40 years ago in discussions with
Professor Zbigniew Oziewicz about the fundamental roll played by duality in its
many different guises in mathematics [4]. The discussion resurfaced recently in
an exchange of emails with Information Scientist Dr. Manfred von Willich. I
hope that my treatment here will open up the discussion to the wider scientific
community.
References
[1] P. De La Harpe, The Clifford algebra and the Spinor group of a Hilbert space,
Compositio Mathematica, tome 25 No. 3, p. 245-261 (1972).
[2] P. Lounesto, Crumeyrolle’s Bivectors and Spinors, in R. Ablamowicz, P.
Lounesto (eds.), Clifford Algebras and Spinor Structures, pp. 137-166, Kluwer
(1995).
[3] L. Narici, On the Hahn-Banach Theorem, p. 87 - 123, in Proceedings of the
Second International Course of Mathematical Analysis in Andalucia, Sept.
20-24, Granada, Editors: M.V. Velasco & Al., World Scientific (2004).
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[4] Z. Oziewicz, From Grassmann to Clifford, p.245-256 in Clifford Algebras
and Their Applications in Mathematical Physics, eds. J.S.R. Chisholm, A.K.
Common, NATO ASI Series C: Mathematical and Physical Sciences Vol. 183
(1986).
[5] D. Shale, W. Stinespring, Spinor Representations of Infinite Orthogonal
Groups, Journal of Mathematics and Mechanics, Vol. 14, No. 2, pp. 315-322
(1965).
[6] G. Sobczyk, Notes on Pl¨uckers relations in geometric algebra, Advances
in Mathematics, 363 (2020) 106959. An early version of this article is
https://arxiv.org/pdf/1809.09706.pdf
[7] G. Sobczyk, Matrix Gateway to Geometric Algebra, Spacetime and Spinors,
Independent Publisher, Nov. 7, 2019.
[8] G. Sobczyk, Periodic Table of Geometric Numbers, 12 March 2020.
https://arxiv.org/pdf/2003.07159.pdf
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