· Geometric Algebra in Linear Algebra and Geometry Jos e Mar a Pozo Departament de F sica...

44
Geometric Algebra in Linear Algebra and Geometry Jos´ e Mar´ ıa Pozo Departament de F´ ısica Fonamental Universitat de Barcelona Diagonal 647, E-08028 Barcelona, Spain jpozo@ffn.ub.es Garret Sobczyk Departamento de Fisica y Matematicas Universidad de las Am´ ericas - Puebla, Mexico 72820 Cholula, M´ exico, [email protected] January 10, 2000, Revised April 15, 2001 Abstract. This article explores the use of geometric algebra in linear and mul- tilinear algebra, and in affine, projective and conformal geometries. Our principal objective is to show how the rich algebraic tools of geometric algebra are fully com- patible with, and augment the more traditional tools of matrix algebra. The novel concept of an h-twistor makes possible a simple new proof of the striking relationship between conformal transformations in a pseudoeuclidean space to isometries in a pseudoeuclidean space of two higher dimensions. The utility of the h-twistor concept, which is a generalization of the idea of a Penrose twistor to a pseudoeuclidean space of arbitrary signature, is amply demonstrated in a new treatment of the Schwarzian derivative. AMS subject classification 15A09, 15A66, 15A75, 17Bxx, 41A10, 51A05, 51A45. Keywords: affine geometry, Clifford algebra, conformal group, euclidean geome- try, geometric algebra, Grassmann algebra, horosphere, Lie algebra, linear algebra, obius transformation, non-Euclidean geometry, null cone, projective geometry, spectral decomposition, Schwarzian derivative, twistor. Contents - 1. Introduction - 2. Geometric Algebra and Matrices nondegenerate geometric algebras spinor basis symmetric and hermitian inner products linear transformations outermorphism and generalized traces characteristic polynomial - 3. Geometric algebra and Non-Euclidean Geometry the meet and joint operations c 2002 Kluwer Academic Publishers. Printed in the Netherlands. gjfinPDF.tex; 6/02/2002; 14:45; p.1

Transcript of  · Geometric Algebra in Linear Algebra and Geometry Jos e Mar a Pozo Departament de F sica...

Page 1:  · Geometric Algebra in Linear Algebra and Geometry Jos e Mar a Pozo Departament de F sica Fonamental Universitat de Barcelona Diagonal 647, E-08028 Barcelona, Spain jpozo@ n.ub.es

Geometric Algebra in Linear Algebra and Geometry

Jose Marıa PozoDepartament de Fısica FonamentalUniversitat de BarcelonaDiagonal 647, E-08028 Barcelona, [email protected]

Garret SobczykDepartamento de Fisica y MatematicasUniversidad de las Americas - Puebla, Mexico72820 Cholula, Mexico,[email protected]

January 10, 2000, Revised April 15, 2001

Abstract. This article explores the use of geometric algebra in linear and mul-tilinear algebra, and in affine, projective and conformal geometries. Our principalobjective is to show how the rich algebraic tools of geometric algebra are fully com-patible with, and augment the more traditional tools of matrix algebra. The novelconcept of an h-twistor makes possible a simple new proof of the striking relationshipbetween conformal transformations in a pseudoeuclidean space to isometries in apseudoeuclidean space of two higher dimensions. The utility of the h-twistor concept,which is a generalization of the idea of a Penrose twistor to a pseudoeuclidean spaceof arbitrary signature, is amply demonstrated in a new treatment of the Schwarzianderivative.

AMS subject classification 15A09, 15A66, 15A75, 17Bxx, 41A10, 51A05, 51A45.

Keywords: affine geometry, Clifford algebra, conformal group, euclidean geome-try, geometric algebra, Grassmann algebra, horosphere, Lie algebra, linear algebra,Mobius transformation, non-Euclidean geometry, null cone, projective geometry,spectral decomposition, Schwarzian derivative, twistor.

Contents

− 1. Introduction

− 2. Geometric Algebra and Matricesnondegenerate geometric algebrasspinor basissymmetric and hermitian inner productslinear transformationsoutermorphism and generalized tracescharacteristic polynomial

− 3. Geometric algebra and Non-Euclidean Geometrythe meet and joint operations

c© 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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2 J. Pozo and G. Sobczyk

affine and projective geometriesexamples

− 4. Conformal Geometrythe horospherethe null coneh-twistorsconformal transformations and isometriesisometries in N0

matrix representationh-twistors and Mobius transformationsthe relative matrix representationconformal transformations in dimension 2

1. Introduction

Almost 125 years after the discovery of “geometric algebra” by WilliamKingdon Clifford in 1878, the discipline still languishes off the center-stage of mathematics. Whereas Clifford’s geometric algebra has gainedcurrency among an increasing number scientists in different “special in-terest” groups, the authors of the present work contend that geometricalgebra should be known by all mathematicians and other scientists forwhat it really is - the natural algebraic completion of the real numbersystem to include the concept of direction. Whereas, evidently, mostmathematicians and other scientists are either unfamiliar with or rejectthis point of view, we will try to prevail by showing that Clifford algebrareally already has been universally recognized in the guise of linearalgebra. Since linear algebra is fully compatible with Clifford algebra, itfollows that in learning linear algebra, every scientist has really learnedClifford algebra but is generally unaware of this fact! What is lackingin the standard treatments of linear algebra is the recognition of thenatural graded structure of linear algebra and, therefore, the geometricinterpretation that goes along with the definition of geometric algebra.As has been often repeated by Hestenes and others, geometric algebrashould be seen as a great unifier of the geometric ideas of mathematics(Hestenes, 1991).

The purpose of the present article is to develop the ideas of geo-metric algebra alongside the more traditional tools of linear algebra bytaking full advantage of their fully compatible structures. There aremany advantages to such an approach. First, everybody knows matrixalgebra, but not everybody is aware that exactly the same algebraicrules apply to the multivectors in a geometric algebra. Because of this

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Geometric Algebra in Linear Algebra and Geometry 3

fact, it is natural to consider matrices whose elements are taken from ageometric algebra. At the same time, by developing geometric algebrain such a way that any problem can be easily changed into an equivalentproblem in matrix algebra, it becomes possible to utilize the powerfuland extensive computer software that has been developed for workingwith matrices. Whereas CLICAL has proven itself to be a powerfulcomputer aid in checking tedious Clifford algebra calculations, it lackssymbolic capabilities (Lounesto, 1994). Geometric algebra offers notonly a comprehensive geometric interpretation but also a whole newset of algebraic tools for dealing with problems in linear algebra. Weshow that matrices, which are rectangular blocks of numbers, repre-sent geometric numbers in a rather special spinor basis of a geometricalgebra with neutral signature.

This work consists of four main chapters. This introductory chapterlays down the rational for this article and gives a brief summary ofits main ideas and content. Chapter 2 is primarily concerned with thedevelopment of the basic ideas of linear and multilinear algebra onan n-dimensional real vector space we call the null space, since we areassuming that all vectors inN are null vectors (the square of each vectoris zero). Taking all linear combinations of sums of products of vectors inN generates the 2n-dimensional associative Grassmann algebra G(N ).This stucture is sufficiently rich to efficiently develop many of the basicnotions of linear algebra, such as the matrix of a linear operator andthe theory of determinants and their properties.

Recently, there has been much interest in the application of geomet-ric algebra to affine, projective and other non-euclidean geometries,(Maks, 1989), (Hestenes, 1991), (Hestenes and Ziegler, 1991), (Porte-ous, 1995) and (Havel, 1995). These noneuclidean models offer newcomputational tools for doing pseudeoeucliean and affine geometryusing geometric algebra. Chapter 3 undertakes a systematic study ofsome of these models, and shows how the tools of geometric algebramake it possible to move freely between them, bringing a unificationto the subject that is otherwise impossible. One of the key ideas is todefine the meet and join operations on equivalence classes of bladesof a geometric algebra which represent subspaces. Since a nonzero r-blade characterizes only the direction of a subspace, the magnitude ofthe blade is unimportant. Basic formulas for incidence relationshipsbetween points, lines, planes, and higher dimensional objects are com-pactly formulated. Examples of calculations are given in the affine planewhich are just plain fun!

Chapter 4 explores the deep relationships which exist between pro-jective geometry and the conformal group. The conformal geometry of apseudo-euclidean space can be linearized by considering the horosphere

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in a pseudo-Euclidean space of two dimensions higher. The introduc-tion of the novel concept of an h-twistor makes possible a simple newproof of the striking relationship between conformal transformationsin a pseudoeuclidean space to isometries in a pseudoeuclidean space oftwo higher dimensions. The concept of an h-twistor greatly simplifiescalculations and is in many ways a generalization of the successfulspinor/twistor formalisms to pseudoeuclidean spaces of arbitrary sig-natures. The utility of the h-twistor concept is amply demonstratedin a new derivation of the Schwarzian derivative (Davis, 1974, p46),(Nehari, 1952, p199).

2. Geometric Algebra and Matrices

Let N be an n-dimensional vector space over a given field K, and let

e = ( e1 e2 · · · en ) (1)

be a basis of N . In this work we only consider real (K = IR) orcomplex (K = IC) vector spaces although other fields could be chosen.By interpreting each of the vectors in e to be the column vectorsof the standard basis of the identity matrix id(n) of the n × n matrixalgebra M(K) over the field K, we are free to make the identificatione = id(n). We wish to emphasize that we are interpreting the basisvectors ei to be elements of the 1 × n row matrix (1), and not theelements of a set. Thus, in what follows, we are assumming and oftenwill apply the rules of matrix multiplication when dealing with the(generalized) row vector of basis vectors e.

Now let N be the dual vector space of 1-forms over the the field K,and let e be the dual basis of N with respect to the basis e of N . Ifwe now interpret each of the vectors in e to be the row vectors of thestandard basis of the identity matrix id(n) of the n×n matrix algebraM(K), we can again make the identification e = id(n). Because wewish to be able to interpret the elements of e as row vectors, we willalways write the vectors in e in the column vector form

e =

e1

e2

··en

(2)

We also assume that the column vector e obeys all the rules of matrixaddition and multiplication of a n× 1 column vector.

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In terms of these bases, any vector or point x ∈ N can be written

x = exe = ( e1 e2 · · en )

x1

x2

··xn

=n∑i=1

xiei (3)

for xi ∈ IR, where

xe =

x1

x2

··xn

are the column vector of components of the vector x with respect tothe basis e.

Since vectors in N are represented by column vectors, and vectorsy ∈ N by row vectors, we define the operation of transpose of the vectorx by

xt = (exe)t = xtee = (x1 x2 . . . xn )

e1

e2

··en

(4)

In the case of the complex field K = C, we have

x∗e = (x1 x2 · · · xn ) =

x1

x2

··xn

t

(5)

The transpose and Hermitian transpose operations allows us to movebetween the reciprocal vector spaces N and N . Clearly the operation ofHermitian transpose reduces to the ordinary transpose for real vectors.

We now wish to weld together the structures of the matrix algebraM(K) and the geometric algebras generated by the vectors in the dualnull spacesN andN . Following (Doran, Hestenes, Sommen, Van Acker,1993), we first consider the Grassmann algebra G(N ), generated bytaking all linear combinations of sums and products of the elements inthe vector space N = spane subject to the condition that for eachx ∈ N , x2 = xx = 0. It follows that

(x+ y)2 = x2 + xy + yx+ y2 = xy + yx = 0 (6)

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or xy = −yx for all x, y in the null space N . The geometric algebraG(N ) generated by a null space N is called the Grassmann or exterioralgebra for the null space N .

As follows from (6), the Grassmann exterior product a1a2 . . . ak ofk vectors in N is antisymmetric over the interchange of any two of itsvectors;

a1 . . . ai . . . aj . . . ak = −a1 . . . aj . . . ai . . . ak

so that the exterior product of null vectors is equivalent to the outerproduct of those vectors:

a1a2 . . . ak = a1∧a2∧ . . .∧ak.

The 2n-dimensional standard basis SBe of G(N ), is generated bytaking all products of the vectors in the standard basis e to getSBe =

1; e1, . . . , en; e12, . . . , e(n−1)n; . . . ; e1···k, . . . , e(n−k+1)···n; . . . ; e12···n =

e0, e1, e2, . . . , en , (7)

where ek := ( e1···k · · · e(n−k+1)···n ) is the(nk

)-dimensional stan-

dard basis of k-vectors

ej1j2···jk ≡ ej1ej2 · · · ejk

for the(nk

)sets of indices 1 ≤ j1 < j2 < · · · < jk ≤ n. In particular, it

is assumed e0 = (1) and e1 = e. The unique component of enis the pseudoscalar or volume element I := e12···n. With respect to thestandard basis SB(e) any multivector X ∈ G(N ) can be expressed inthe matrix form

X = SBeXSB (8)

where XSB is the column vector of components

XSB =

x0xexe2··

xen

Just as we used the tranpose operation (4) to move from the the

null space N = spane to the dual null space N , we can extend thedefinition of the transpose to enable us to move from the Grassmannalgebra G(N ), to the Grassmann algebra G(N ) of the reciprocal null

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spaceN . Since multivectors in G(N ) are represented by column vectors,and multivectors Y ∈ G(N ) by row vectors, we define the transposeXt ∈ G(N ) by

Xt = (SBeXSB)t = XtSBSBe (9)

where XtSB is the row vector of components

XtSB =

(xt0 xte xte2

· · xten).

The Hermitian transpose is similarly defined when we are dealing withcomplex multivectors.

The dual basis of multivectors SBe for G(N ) are arranged in acolumn and are defined by

SBe = (SBe)t =

1ee2··en

where

ek :=

ek...1··

en...n−k+1

(10)

is the(nk

)-dimensional basis of dual k-vectors defined by

ej1j2...jk ≡ ej1ej2 · · · ejk

for the(nk

)sets of indices n ≥ j1 > j2 > . . . > jk ≥ 1.

The dual space N of the space N , and more generally the dualGrassmann algebra G(N ) of the Grassmann algebra G(N ), are definedto satisfy the usual properties of the mathematical dual space. What(Doran, Hestenes, Sommen, Van Acker, 1993) observed was that thesesame properties can be faithfully expressed in a larger neutral geometricalgebra Gn,n (a fomal definition is given below) containing both of theseGrassmann algebras as subalgebras, by replacing the duality conditionswith corresponding reciprocal conditions. We accomplish all this byassuming the additional properties

e2i = 0 = e2

i , eiej = −ejei, eiej = −ejei (for i 6= j), and eiej = −ejei,(11)

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8 J. Pozo and G. Sobczyk

together with the reciprocal relations

ei · ej = δi,j = ej · ei (12)

for all i, j = 1, 2, . . . , n. With this definition, the Grassmann alge-bra G(N ) of the dual space N becomes the natural reciprocal of theGrassmann algebra G(N ). These relations imply that the reciprocal k-vectors and k-forms of Grassmann algebras G(N ) and G(N ) satisfy thereciprocal relations

ek · ek = id((nk

)×(nk

)) .

The neutral pseudoeuclidean space IRn,n is defined as the linear spacewhich contains both the null spaces N and N . Thus,

IRn,n = N ⊕N = x+ y| x ∈ N , y ∈ N.

Likewise, the 22n–dimensional associative geometric algebra Gn,n is de-fined to be the geometric algebra that contains both the Grassmannalgebras G(N ) and G(N ). We write

Gn,n = G(N )⊗ G(N ) = gene1, ..., en, e1, ..., en, (13)

subjected to the relationships (11) and (12).A simple example will serve to show the interplay between the well-

known matrix multiplication and the geometric product in the supermatrix algebra M(Gn,n). Recalling the basic geometric product of twovectors x, y,

xy = x · y + x∧y, (14)

we apply the same product to the of row and column basis vectors eand e, and simultaneously employ matrix multiplication, to get theexpressions

ee = e·e+e∧e = id(n×n)+

e1∧e1 e1∧e2 . . . e1∧ene2∧e1 e2∧e2 . . . e2∧en. . . . . . . . . . . .. . . . . . . . . . . .

en∧e1 en∧e2 . . . en∧en

where id(n × n) is the n × n identity matrix, computed by taking allinner products ei ·ej between the basis vectors of e and e. Similarly,

ee = e · e+ e∧e =n∑i=1

ei · ei +n∑i=1

ei∧ei = n+n∑i=1

ei∧ei,

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Geometric Algebra in Linear Algebra and Geometry 9

giving the useful formulas

e · e = n and e∧e =n∑i=1

ei∧ei (15)

Because of the metrical structure induced by the reciprocal relation-ships (12), we can express the components xe of the vector x ∈ N (3)in the form

xe =

x1

x2

··xn

=

e1 · xe2 · x··

en · x

= e · x

Similarly, the components of the reciprocal vector xt ∈ N can be foundfrom

xte = (x1 x2 . . . xn ) = xt · ( e1 e2 · · en ) (16)

We call Gn,n the universal geometric algebra of order 22n. When n iscountably infinite, we call G = G∞,∞ the universal geometric algebra.The universal algebra G contains all of the algebras Gn,n as propersubalgebras. In (Doran, Hestenes, Sommen, Van Acker, 1993), Gn,n iscalled the mother algebra.

2.1. nondegenerate geometric algebras

The standard bases e and e of the reciprocal null spaces N and N ,taken together, are said to make up a Witt basis of null vectors (Ablam-owicz and Salingaros, 1985) of the neutral pseudoeuclidean space IRn,n.From the Witt basis, we can construct the standard orthonormal basisof IRn,n σ, η of Gn,n,

σi = ei +12ei ηi = ei −

12ei (17)

for i = 1, 2, . . . , n. Using the defining relationships (12) of the reciprocalframes e and e, we find that these basis vectors satisfy

σi2 = 1 ηi

2 = −1 ηiσj = −σjηi ∀ i, j = 1, . . . , n

σiσj = −σjσi ηiηj = −ηjηi ∀ i 6= j

The basis σ spans a real Euclidean vector space IRn and generatesthe geometric subalgebra Gn,0, whereas η spans an anti–Euclidean

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10 J. Pozo and G. Sobczyk

space IR0,n and generates the geometric subalgebra G0,n. The standardbases (7) of these geometric algebras naturally take the forms

SBσ and SBη,

so that a general multivector X ∈ Gn,0 can be written

X = SBσXSB

and similarly for an X ∈ Gn,0. We can now express the geometricalgebra Gn,n as the product of these geometric subalgebras

Gn,n = Gn,0 ⊗ G0,n = genσ1, ..., σn, η1, ..., ηn , (18)

again only as linear spaces, but not as algebras.Notice that when we write down the relationship (17), we have given

up the possibility of interpreting the vectors in e and e as columnand row vectors, respectively. When working in the nondegenerate ge-ometric algebras Gn,n, Gn,0 or G0,n, we use the operation of reversal.The reversal of any vector x ∈ Gn,n is defined by x† := x, and for thek-vector Ak = a1∧a2∧ . . .∧ak,

Ak† := ak∧ak−1∧ . . .∧a1 = (−1)k(k−1)/2Ak.

2.2. Spinor basis

One nice application of the above formalism is that it allows us tosimply express a natural isomorphism that exists between the neutralgeometric algebra Gn,n, and the algebra of all real 2n × 2n matricesMIR(2n). To express this isomorphism, we first define 2n mutuallycommuting idempotents

ui(±) =12

(1± σiηi) (19)

for i = 1, 2, . . . , n.We can now define 2n mutually annihiliating primitive idempotents

for the algebra Gn,n,

usigns =∏signs

ui(signsi) (20)

where signs is a particular sequence of n ± signs, and signsi is the ith

sign in the sequence. For example,

u+++...+ =n∏i=1

ui(+) and u−−−...− =n∏i=1

ui(−).

The primitive idempotents satisfy the following basic properties

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Geometric Algebra in Linear Algebra and Geometry 11

−∑2n

signs usigns = 1

− σiu+++...+ = u+...+−i+...+σi

− usign1usign2 = δsign1 sign2usign1 where δsign1 sign2 = 0

except when sign1 = sign2 for which δsign1 sign2 = 1.

The above properties are easily verified.In contrast to the standard basis SBe of the neutral geometric

algebra Gn,n, the spinor basis of Gn,n is defined to be the 2n × 2n

multivectors in the matrix

SNB(n, n) = SBσtu+++...+SBσ (21)

The simplest example is the spinor basis for the geometric algebraG1,1. The 21 primitive idempotents for this geometric algebra are u± =12(1± ση). Using (21), the spinor basis SNB(1, 1) is found to be

SB(σ)tu+SB(σ) =(

)u+ ( 1 σ ) =

(u+ σu−σu+ u−

)The significance of the position of each multivector in the spinor basis,is that its matrix representation corresponds to a 1 in the same position(with zeros everywhere else).

In terms of the spinor basis, any 2n × 2n matrix A represents thecorresponding element A ∈ Gn,n given by

A = SBσu+++...+A SBσt.

The matrix A associated with the multivector A ∈ Gn,n is denoted byA = [A]. This association constitutes an algebra isomorphism, since[A+B] = [A] + [B] and [AB] = [A][B]. Noting that

u+···+ SBσtSBσu+···+ = u+···+ id(2n × 2n),

it easily follows that

AB = SBσu+···+ [A]SBσt SBσu+···+ [B]SBσt

= SBσu+···+ [A][B]SBσt (22)

We will use the spinor basis SNB(1, 1) for studying conformal trans-formation in section 4.

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12 J. Pozo and G. Sobczyk

2.3. Symmetric and Hermitian Inner Products

Until now we have only considered real geometric algebras and theircorresponding real matrices. Any pseudoscalar of the geometric algebraGn,n will always have a positive square, and will anticommute withthe vectors in IRn,n. If we insisted on dealing only with real geometricalgebras, we might consider working in the geometric algebra Gn,n+1

where the pseudoscalar element i has the desired property that i2 = −1and is in the center of the algebra (commutes with all multivectors). Acomplex vector x+ iy in Gn,n+1 consists of the real vector part x and apseudovector or (2n)-blade part iy.

Instead, we choose to directly complexify the geometric algebra Gn,nto get the complex geometric algebra G2n(IC) (Sobczyk, 1996). Whereasthis algebra is isomorphic to Gn,n+1, it is somewhat easier to work withthan the former. A complex vector z ∈ IC2n has the form z = x + iywhere x, y ∈ IR2n. The imaginary unit i, where i2 = −1, is defined tocommute with all elements in the geometric algebra G2n(IC).

Consider an orthonormal basis σ ∈ IC2n : σi · σj = δij . Thecomplexified null space N (IC) and its reciprocal null space N (IC) arethe subspaces spanned by the complex null vectors

ej =12

(σj + iσn+j) and ej = σj − iσn+j

for j = 1, 2, . . . , n. This definition is consistent with (17) if we considerηj = iσn+j . Thus a null vector x ∈ N (IC) has the form x = exe forxi ∈ IC.

Previously we have defined the transposition (4). This operationcan be extended to complex vectors in two different ways. The firstway is a linear extension. We use the term transposition for the linearextension, so that the definition (4) is still valid when xi ∈ IC. Thesecond extension is antilinear and is equivalent to Hermitian conju-gation: x∗ ≡ x∗ee. Both operations, Hermitian conjugation andtransposition, take us from the complex null space N (IC) to the dualnull space N (IC), and if the components of x are all real, both reduceto the real transposition. Applied to the components xe, x∗e is theusual Hermitian transpose of the column vector xe,

x∗e = (x1 x2 · · · xn ) =

x1

x2

··xn

t

(23)

We now define the symmetric inner product (x, y), and the Hermi-tian inner product 〈x, y〉, on N (IC). For all x, y ∈ N (IC), the two prod-

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Geometric Algebra in Linear Algebra and Geometry 13

ucts are defined, respectively, by using transposition and Hermitionconjugation:

(x, y) := xt · y = xteye and 〈x, y〉 := x∗ · y = x∗eye (24)

The Hermitian inner product will be used in the next subsection.

2.4. Linear Transformations

Let N ⊕N ′ and N ⊕N ′ be (n+n′)-dimensional reciprocal null spacesin IRn+n′,n+n′ with the dual bases e ∪ e′ and e ∪ e′. Let f :N → N ′ be a linear transformation from the null space N into thenull space N ′. In light of the previous section, we can consider the nullspaces N and N ′ to be over the real or complex numbers. Let

Hom(N ,N ′) = f : N → N ′| f is a linear transformation

denote the linear space of all homomorphisms from N to N ′, with theusual operation of addition of transformations. Of course, only whenN = N ′ is the operation of multiplication (composition) defined.

Given an operator f ∈ Hom(N ,N ′), y′ = f(x) ≡ fx, the matrix Fof f with respect to the bases e and e′ is defined by

fe ≡ ( fe1 · · · fen ) = ( e1 · · · e′n′ )F = e′F . (25)

Of course, the matrix F = (fij) is defined by its n′ × n componentsfij = ei · f(ej) ∈ C for i = 1, 2, . . . , n′ and j = 1, 2, . . . , n. It followsthat f(ej) =

∑n′

i=1 e′ifij . By dotting both sides of the above equation

on the left by e′, we find the explicit expression

F = e′ · e′F = e′ · fe .

Equation (15) can be used to define the bivector F of the linearoperator f . It is defined by

F = fe∧e′

and satisfies the property that f x = F · x for all x ∈ N . The bivectorof a linear operator makes possible a new theory of linear operators,and is particularly useful in defining the general linear group as a Liegroup of bivectors with the commutator product, (Fulton and Harris,1991), (Eds. Bayro and Sobczyk, 2001, pp. 32).

Given the Hermitian inner product (24), the transpose (or Hermitiantranspose (23)) f∗ : N ′ → N of the mapping f : N → N ′ is defined bythe requirement that for all x ∈ N and y′ ∈ N ′,

〈x, f∗(y′)〉 = 〈f(x), y′〉 ⇔ F∗ ≡ e · f∗e′ = [fe]∗ · e′ .

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14 J. Pozo and G. Sobczyk

Likewise, we can define the transpose relative to the symmetric innerproduct

(x, f t(y′)) = (f(x), y′) ⇔ F t ≡ e · f te′ = [fe]t · e′ .

2.5. Outermorphism and generalized traces

A linear transformation f naturally extends multilinearly to act onk-blades,

f(x∧y∧ · · · ∧z) ≡ f(x)∧f(y)∧ · · · ∧f(z) ∀x, y, . . . , z ∈ N ,

and where f(1) ≡ 1. Thus extended, f : G(N ) → G(N ′) is calledthe outermorphism of the linear transformation f : N → N ′, since itpreserves the structure of the outer product:

f(A∧B + C∧D) = f(A)∧f(B) + f(C)∧f(D) ∀A,B,C,D ∈ G(N ) .

Geometrically, the outermorphism f maps directed areas into directedareas, and more generally, directed k-vectors into directed k-vectors.

A linear transformation from N into itself is called an endomor-phism. Let

End(N ) = f : N → N| f is a linear operator

denote the algebra of all endomorphisms on N . The operations ofaddition and composition of linear operators is well defined for en-domorphisms.

The determinant det f of the endomorphism f is defined to be theeigenvalue of the pseudoscalar element I = e12···n:

f(I) = det f I ⇔ det f = f(I) · I

Thus, det f is the factor by which volume is scaled by f . The trace of fis defined by trf := fe ·e. Given the outermorphism of f we definethe generalized traces of f by

trif := fei · ei .

Particular cases are tr0f = f(1) ·1 = 1 and tr1f = trf . The generalizedtrace of degree n coincides with the determinant: trnf = f(I)·I = det f .

A second basis a of N is related to the standard basis e by theapplication of some endomorphism a

a = ae = eA = ( e1 e2 · · · en )A (26)

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Geometric Algebra in Linear Algebra and Geometry 15

where A is called the matrix of transition from the basis e to thebasis a. Taking the outer product

∧ni=1a of the basis vectors a,

we getn∧i=1

a ≡ a1∧a2∧ · · · ∧an = a(e1∧e2∧ · · · ∧en) = det a I . (27)

We see from (27) that the determinant of the matrix of transition,detA ≡ det a, between two bases cannot be zero.

We can now easily construct a dual or reciprocal basis a for thebasis a :

ai = (−1)i+1 (a1∧ . . .∧i∗∧ . . .∧an) · Idet a

(28)

where i∗ means that ai is omitted from the product. More compactly,using our matrix notation,

a =a(e · I) · Ia(I) · I

.

Checking, we find that

a · a =[ a(e·I ) · I ] · a

a(I) · I=

[ a( I ·e)∧ae ] · Ia(I) · I

=[ a( ( I ·e)∧e) ] · I

a(I) · I=a(I (e·e) ) · I

a(I) · I= e · e = id

We have actually found the inverse of the transition matrix A, givenby A−1 = a · e, (Eds. Bayro and Sobczyk, 2001, p.25).

2.6. characteristic polynomial

The characteristic polynomial of f : N → N is defined by

ϕf (λ) = det(λ− f) = (λ− f)(I) · I.

The well-known Caley-Hamilton theorem, which says that every linearoperator satisfies its characteristic equation, is a consequence of theidentity

f [x∧en−1] · en−1 = (x∧en−1) · en−1det f = xdet f (29)

When the left side of this identity is expanded we get

f [x∧en−1] · en−1 =n∑i=1

(−1)i+1fen−i · en−i fi(x)

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16 J. Pozo and G. Sobczyk

=n∑i=1

(−1)i+1 trn−if f i(x) . (30)

Expressed in terms of the generalized traces of f , the characteristicpolynomial is

ϕf (λ) = (λ− f)(e12···n) · en···21 =n∑i=0

(−1)ifei · eiλn−i.

Thus, from (29) and (30), we have ϕf (f) = 0, i.e. f satisfies its char-acteristic polynomial.

The above equation (29) can also be used to derive a formula forthe inverse of f . We get

x = f−1(y) =(y∧fen−1) · en−1

det f.

The minimal polynomial ψf (λ) of f is the polynomial of least degreethat has the property that ψf (f) = 0. Taken over the complex numbersIC, we can express ϕf and ψf in the factored form

ϕf (λ) =r∏i=1

(λ− λi)ni and ψf (λ) =r∏i=1

(λ− λi)mi

where 1 ≤ mi ≤ ni ≤ n for i = 1, 2, . . . , r, and the roots λi are alldistinct.

The minimal polynomial uniquely determines, up to an ordering ofthe idempotents, the following spectral decomposition theorem of thelinear operator f , (Sobczyk, 2001).

THEOREM 1. If f has the minimal polynomial ψ(λ), then a set ofcommuting mutually annihilating idempotents and corresponding nilpo-tents (pi, qi)| i = 1, . . . , r can be found such that

f =r∑i=1

(λi + qi)pi,

where rank(pi) = ni, and the index of nilpotency index(qi) = mi, fori = 1, 2, . . . , r. Furthermore, when mi = 1, qi = 0.

Clearly, the operator f is diagonalizable if and only if it has the spectralform

f =r∑i=1

λipi.

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The spectral decomposition theorem has many different uses and ap-plies equally well to a linear operator or a geometric number, (Sobczyk,1993, pp357-364), (Sobczyk, 1997; Sobczyk, 1997a). For example, wecan immediately define a generalized inverse of the operator f by

f inv =∑λi 6=0

1λi

(pi −qiλi

+ · · ·+ (−qiλi

)mi−1)

satisfying the conditions ff inv = f invf =∑

λi 6=0 pi, (Rao and Mitra,1971, pp.20).

3. Geometric algebra and Non-Euclidean Geometry

Leonardo da Vinci (1452-1519) was one of the first to consider theproblems of projective geometry. However, projective geometry was notformally developed until the work “Traite des propries projectives desfigure” of the French mathematician Poncelet (1788-1867), publishedin 1822. The extrordinary generality and simplicity of projective ge-ometry led the English mathematician Cayley to exclaim: “ProjectiveGeometry is all of geometry” (Young, 1930).

Let IRn+1 be an (n+ 1)-dimensional euclidean space and let Gn+1,0

be the corresponding geometric algebra. The directions or rays of non-zero vectors in IRn+1 are identified with the points of the n-dimensionalprojective plane Πn, (Hestenes and Ziegler, 1991). More precisely, wewrite

Πn ≡ IRn+1/IR∗

where IR∗ = IR−0. We thus identify points, lines, planes, and higherdimensional k-planes in Πn with 1, 2, 3, and (k + 1)-dimensional sub-spaces Sr of IRn+1, where k ≤ n. To effectively apply the tools ofgeometric algebra, we need to introduce the new basic operations ofmeet and join, (Eds. Bayro and Sobczyk, 2001, p.27).

3.1. The Meet and Joint Operations

The meet and join operations of projective geometry are most easilydefined in terms of the intersection and direct sum of the subspaceswhich name the objects in Πn. On the other hand, each r-dimensionalsubspace Ar can be described by a non-zero r-blade Ar ∈ G(IRn+1). Wesay that an r-blade Ar represents, or is a representant of an r-subspaceAr of IRn+1 if and only if

Ar = x ∈ IRn+1| x∧Ar = 0. (31)

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18 J. Pozo and G. Sobczyk

We denote the equivalence class of all nonzero r-blades Ar ∈ G(IRn+1)which define the subspace Ar by

Array := tAr | t ∈ IR, t 6= 0. (32)

Evidently, every r-blade in Array is a representant of the subspaceAr. With these definitions, the problem of finding the meet and join isreduced to a problem in geometric algebra of finding the correspondingmeet and join of the (r+1)- and (s+1)-blades in the geometric algebraG(IRn+1) which represent these subspaces.

Let Ar, Bs and Ct be non-zero blades representing the three sub-spaces Ar, Bs and Ct, respectively. We say that

DEFINITION 1. The t-blade Ct = Ar ∩ Bs is the meet of Ar and Bsif there exists a complementary (r − t)-blade Ac and a complementary(s− t)-blade Bc with the property that Ar = Ac∧Ct, Bs = Ct∧Bc, andAc∧Bc 6= 0.

It is important to note that the t-blade Ct ∈ Ctray is not uniqueand is defined only up to a non-zero scalar factor, which we chooseat our own convenience. The existence of the t-blade Ct (and thecorresponding complementary blades Ac and Bc) is an expression ofthe basic relationships that exists between subspaces.

DEFINITION 1.1. The (r + s− t)-blade D = Ar ∪Bs, called the joinof Ar and Bs is defined by D = Ar ∪Bs = Ar∧Bc.

Alternatively, since the join Ar ∪Bs is defined only up to a non-zeroscalar factor, we could equally well define D by D = Ac∧Bs. We usethe symbols ∩ intersection and ∪ direct sum from set theory to markthis unusual state of affairs. The problem of “meet” and “join” hasthus been solved by finding the direct sum and intersection of linearsubspaces and their (r + s− t)-blade and t-blade representants.

Note that it is only in the special case when Ar ∩ Bs = 0 that thejoin can be considered to reduce to the outer product. That is

Ar ∩Bs = 0 ⇔ Ar ∪Bs = Ar∧Bs

However, after the join IAr∪Bs ≡ Ar ∪ Bs has been found, it can beused to find the meet Ar ∩Bs,

Ar ∩Bs = Ar · [Bs · IAr∪Bs ] = [IAr∪Bs ·Ar] ·Bs (33)

While the positive definite metric of IRn+1 is irrelevant to the definitionof the meet and join of subspaces, the formula (33) holds only in IRn+1.

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Geometric Algebra in Linear Algebra and Geometry 19

A slightly modified version of this formula will hold in any non-degenerate pseudoeuclidean space IRp,q, where p+q = n+1. In this case,after we have found the join IAr∪Bs , which is a (r+ k)-blade, we find areciprocal (r+k)-blade IAr∪Bs with the property that IAr∪Bs ·IAr∪Bs 6=0. The meet Ar ∩Bs may then be defined by

Ar ∩Bs = Ar · [Bs · IAr∪Bs ] = [IAr∪Bs ·Ar] ·Bs (34)

3.2. affine and projective geometries

We have seen in the previous section how the meet and join of the n di-mensional projective space Πn can be defined in an (n+1)-dimensionaleuclidean space IRn+1. There is a very close connection between affineand projective geometries. A projective space can be considered to bean affine space with idealized points at infinity (Young, 1930). Since allthe formulas for meet and join remain valid in the pseudoeuclideanspace IRp,q, subject only to (34), we will define the n = (p + q)-dimensional affine plane Ae(IRp,q) of the null vector e = 1

2(σ + η)in the larger pseudoeuclidean space IRp+1,q+1 = IRp,q ⊕ IR1,1, whereIR1,1 = spanσ, η for σ2 = 1 = −η2. Whereas, effectively, we are onlyextending the euclidean space IRp,q by the null vector e, it is advanta-geous to work in the geometric algebra Gp+1,q+1 of the non-degeneratepseudoeuclidean space IRp+1,q+1.

The affine plane Ap,qe := Ae(IRp,q) is defined by

Ae(IRp,q) = xh = x+ e| x ∈ IRp,q ⊂ IRp+1,q+1, (35)

for the null vector e ∈ IR1,1. The affine plane Ae(IRp,q) has the niceproperty that x2

h = x2 for all xh ∈ Ae(IRp,q), thus preserving the metricstructure of IRp,q. By employing the reciprocal null vector e = σ−η withthe property that e·e = 1, we can restate definition (35) of Ae(IRp,q) inthe form

Ae(IRp,q) = y| y ∈ IRp+1,q+1 , y·e = 1 and y·e = 0 ⊂ IRp+1,q+1

This form of the definition is interesting because it brings us closer tothe definition of the n = (p+ q)-dimensional projective plane .

We summarize here the important properties of the reciprocal nullvectors e = 1

2(σ+ η) and e = σ− η that will be needed later, and theirrelationship to the hyperbolic unit bivector u := ση.

e2 = e2 = 0, e·e = 1, u = e∧e = σ∧η, u2 = 1 (36)

The projective n-plane Πn can be defined to be the set of all pointsof the affine plane Ae(IRp,q), taken together with idealized points at

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20 J. Pozo and G. Sobczyk

infinity. Each point xh ∈ Ae(IRp,q) is called a homogeneous representantof the corresponding point in Πn because it satisfies the property thatxh ·e = 1. To bring these different viewpoints closer together, points inthe affine plane Ae(IRp,q) will also be represented by rays in the space

Arayse (IRp,q) = yray| y ∈ IRp+1,q+1, y ·e = 0, y ·e 6= 0 ⊂ IRp+1,q+1

(37)The set of rays Arayse (IRp,q) gives another definition of the affine n-plane, because each ray yray ∈ Arayse (IRp,q) determines the uniquehomogeneous point

yh =y

y ·e∈ Ae(IRp,q).

Conversely, each point y ∈ Ae(IRp,q) determines a unique ray yray inArayse (IRp,q). Thus, the affine plane of homogeneous points Ae(IRp,q) isequivalent to the affine plane of rays Arayse (IRp,q).

Suppose that we are given that we are given k-points ah1 , ah2 , . . . , a

hk ∈

Ae(IRp,q) where each ahi = ai+e for ai ∈ IRp,q. Taking the outer productor join of these points gives the projective (k − 1)-plane Ah ∈ Πn.Expanding the outer product gives

Ah = ah1∧ah2∧ . . .∧ahk = ah1∧(ah2 − ah1)∧ah3∧ . . .∧ahk

= ah1∧(ah2 − ah1)∧(ah3 − ah2)∧ah4∧ . . .∧ahk = . . .

= ah1∧(a2 − a1)∧(a3 − a2)∧ . . .∧(ak − ak−1),

orAh = ah1∧ah2∧ . . .∧ahk = a1∧a2∧ . . .∧ak+e∧(a2 − a1)∧(a3 − a2)∧ . . .∧(ak − ak−1). (38)

Whereas (38) represents a (k−1)-plane in Πn, it also belongs to theaffine (p, q)-plane Ap,qe , and thus contains important metrical informa-tion. Dotting this equation with e, we find that

e·Ah = e·(ah1∧ah2∧ . . .∧ahk) = (a2 − a1)∧(a3 − a2)∧ . . .∧(ak − ak−1).

This result motivates the following

DEFINITION 1.1.1. The directed content of the (k− 1)-simplex Ah =ah1∧ah2∧ . . .∧ahk in the affine (p, q)-plane is given by

e·Ah

(k − 1)!=e·(ah1∧ah2∧ . . .∧ahk)

(k − 1)!

=(a2 − a1)∧(a3 − a2)∧ . . .∧(ak − ak−1)

(k − 1)!

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Geometric Algebra in Linear Algebra and Geometry 21

3.3. examples

Many incidence relations can be expressed in the affine plane Ae(IRp,q)which are also valid in the projective plane Πn, (Eds. Bayro and Sobczyk,2001, pp.263). A few examples are provided below.

Given 4 coplanar points ah, bh, ch, dh ∈ Ae(IR2). The join and meetof the lines ah∧bh and ch∧dh are given, respectively, by (ah∧bh) ∪(ch∧dh) = ah∧bh∧ch, and using (34)

(ah∧bh) ∩ (ch∧dh) = [I ·(ah∧bh)]·(ch∧dh)

where I = σ2∧σ1∧e. Carrying out the calculations for the meet andjoin, we find that

(ah∧bh) ∪ (ch∧dh) = detah, bh, chI = deta, bI (39)

where I = σ1∧σ2∧e, and

(ah∧bh) ∩ (ch∧dh) = detc− d, b− cah + detc− d, c− abh (40)

Note that the meet (40) is not, in general, a homogeneous point.Normalizing (40), we find the homogeneous point ph ∈ Ae(IR2)

ph =detc− d, b− cah + detc− d, c− abh

detc− d, b− a

which is the intersection of the lines ah∧bh and ch∧dh, see Figure 1.The meet can also be solved for directly in the affine plane by notingthat

ph = αpah + (1− αp)bh = βpch + (1− βp)dhand solving to get αp = detbh, ch, dh/detbh − ah, ch, dh.

Given the line ah∧bh ∈ Ae(IR2) and a third point dh ∈ Ae(IR2), as inFigure 1, the point fh on the line ah∧bh which is closest to the point dhis called the foot of the point dh on the line ah∧bh. Since fh∧ah∧bh = 0,it follows that fh = αfah + (1− αf )bh and fh∧bh = αfah∧bh. We cansolve this last equation for αf by dotting it with e, and invoking theauxilliary condition that (b− f) · (d− f) = 0. We get

αf =(a− b).(d− b)

(a− b)2(41)

It should be carefully noted that ah − bh = a − b ∈ IR2 for any twohomogeneous points ah, bh ∈ A2

e. It follows that the foot fh on the lineah∧bh is given by

fh =(b− d)·(b− a)ah + (a− d)·(a− b)bh

(a− b)2. (42)

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22 J. Pozo and G. Sobczyk

Saying that ah, bh, ch ∈ A2e are non-collinear points is equivalent

to the condition ah∧bh∧ch 6= 0. If dh is any other point in A2e, then

dh∧ah∧bh∧ch = 0 so that

dh = αdah + βdbh + (1− αd − βd)ch.

By wedging this last equation by bh∧ch and ah∧ch, respectively, we caneasily solve for αd and βd, getting

αd =detdh, bh, chdetah, bh, ch

and βd =detdh, ch, ahdetah, bh, ch

(43)

ph

ch

bh

d

f

a

h

h

h

Figure 1. Incidence relationships in the affine plane.

Three non-collinear points ah, bh, ch ∈ A2e determine a unique circle

with center rh = αrah + βrbh + (1 − αr − βr)ch. To find the center,note that rh lies on the intersection of the perperdicular bisectors ofthe cords ah∧bh and ah∧ch, and therefore satisfies

rh =12

(ah + bh) + s(ch − wh) =12

(ah + ch) + t(bh − qh), (44)

where

wh = fwah + (1− fw)bh, and qh = fqah + (1− fq)chare the feet (42) of ch and bh along the lines ah∧bh and ah∧ch, respec-tively, for

fw =(a− b·(c− b)

(a− b)2)and fq =

(c− a)·(b− a)(c− a)2)

.

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Geometric Algebra in Linear Algebra and Geometry 23

From (44), it follows that

(fws− fqt)ah + [t+ (1− fw)s− 12

]bh + [12− (1− fq)t− s]ch ≡ 0,

which gives

s =fq

2fq(1− fw) + 2fwand t =

fw2fq(1− fw) + 2fw

.

After simplification, the center rh is found to be

rh =[fq + fw − 2fbfw]ah + fwbh + fbch

2[fq + fw − fbfw]. (45)

Another theorem of interest is Simpson’s theorem for the circle. Wehave assembled all of the tools necessary for a proof of this venerabletheorem in the affine plane Ae(IR2), but we will not prove it here(Eds. Bayro and Sobczyk, 2001, pp.39). Simpson’s theorem has alsobeen proven in the non-linear horosphere (Li, Hestenes, and Rockwood,2000), but the proof is not trivial. It remains to be seen if there are anyreal advantages to proving such theorems on the horosphere and notin the simpler affine plane. The issue at hand is how to best representproblems in distance geometry (Dress and Havel, 1993).

Hestenes and Zigler have also given a proof of Desargues theoremin the projective plane Π2 (Hestenes and Ziegler, 1991), by using itsrepresentation in the euclidean space IR3. A proof of Desargues theoremcan also be given in the affine plane of rays Arayse (IRp,q), (Eds. Bayroand Sobczyk, 2001, pp.37). The importance of such proofs is that eventhough geometric algebra is endowed with a metric, there is no reasonwhy we cannot use the tools of euclidean space to give a proof of thismetric independent result. Indeed, as has been emphasized by Hestenesand others (Barnabei, Brini, and Rota, 1985), all the results of linearalgebra can be supplied with such a projective interpretation.

4. Conformal Geometry

The conformal geometry of a pseudo-Euclidean space can be linearizedby considering the horosphere in a pseudo-Euclidean space of two di-mensions higher. Because it is so easy to introduce extra orthogonalanticommuting vectors into a geometric algebra, without altering thestructure of the geometric algebra in any other way, the framework ofgeometric algebra offers a unification to the subject that is impossible inother formalisms. The horosphere has recently attracted the attention

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24 J. Pozo and G. Sobczyk

of many workers, see for example, (Dress and Havel, 1993; Porteous,1995; Havel, 1995).

The horosphere and null cone are formally introduced in subsections4.1 and 4.2. In subsection 4.3, the concept of an h-twistor is introducedwhich will greatly simplify computations. An h-twistor is a generaliza-tion of the Penrose twistor concept. In subsection 4.4, we give a simpleproof, using only basic concepts from differential geometry developedin (Hestenes and Sobczyk, 1984), of an intriging result that relatesconformal transformations in a pseudoeuclidean space to isometries ina pseudoeuclidean space of two higher dimensions. The original proof ofthis striking relationship was given by (Haantjes, 1937). In subsection4.5, we show that for any dimension greater than two, that any isometryon the null cone can be extended to all of the pseudoeuclidean space.

In subsections 4.6 and 4.7, we show the beautiful relationships thatexists between Mobius transformations (linear fractional transforma-tions) and their 2 × 2 matrix representation over a suitable geometricalgebra. In a final subsection, we explore how all of the formalism devel-oped in the previous sections can be utilized in the characterization ofconformal transformations of the pseudoeuclidean space IRp,q. We de-velop the theory in a novel way which suggests a non-trivial generaliza-tion of the theory of two-component spinors and 4-component twistors.Recall that a conformal transformation preserves angles between tan-gent vectors at each point (Lounesto and Springer, 1989; Porteous,1995) . The utility of the h-twistor concept is amply demonstrated ina new derivation of the Schwarzian derivative.

We begin by defining the horosphere Hp,qe in IRp+1,q+1 by moving upfrom the affine plane Ap,qe := Ae(IRp,q).

4.1. the horosphere

Let Gp+1,q+1 = gen(IRp+1,q+1) be the geometric algebra of IRp+1,q+1,and recall the definition (35) of the affine plane Ap,qe := Ae(IRp,q) ⊂IRp+1,q+1. Any point y ∈ IRp+1,q+1 can be written in the form y =x+ αe+ βe, where x ∈ IRp,q and α, β ∈ IR.

The horosphere Hp,qe is most directly defined by

Hp,qe := xc = xh + βe | xh ∈ Ap,qe and x2c = 0. (46)

With the help of (36), the condition that

x2c = (xh + βe)2 = x2 + 2β = 0

gives us immediately that β := −x2

2 . Thus each point xc ∈ Hp,qe has theform

xc = xh −x2h

2e = x+ e− x2

2e =

12xhexh. (47)

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Geometric Algebra in Linear Algebra and Geometry 25

The last equality on the right follows from

12xhexh =

12

[(xh ·e)xh + (xh∧e)xh] = xh −12x2he.

Just as xh ∈ Ap,qe is called the homogeneous representant of x ∈ IRp,q,the point xc is called the conformal representant of both the pointsxh ∈ Ap,qe and x ∈ IRp,q. The set of all conformal representants Hp,q :=c(IRp,q) is called the horosphere . The horosphere Hp,q is a non-linearmodel of both the affine plane Ap,qe and the pseudoeuclidean space IRp,q.The horosphere Hn for the Euclidean space IRn was first introducedby F.A. Wachter, a student of Gauss, (Havel, 1995), and has beenrecently finding many diverse applications (Eds. Bayro and Sobczyk,2001, chapter 1, chapter 4, chapter 6).

Defining the bivector Kx := e∧xc = e∧xh, it is easy to get back xhby the simple projection,

xh = e·Kx (48)

and to x ∈ IRp,q, by

x = u·(u∧xc) = e·(e∧xh), (49)

using the bivector u defined in (36).The set of all null vectors y ∈ IRp+1,q+1 make up the null cone

N := y ∈ IRp+1,q+1| y2 = 0.

The subset of N containing all the representants y ∈ xcray for anyx ∈ IRp,q is defined to be the set

N0 = y ∈ N | y ·e 6= 0 = ∪x∈IRp,q xcray,

and is called the restricted null cone. The conformal representant of anull ray zray is the representant y ∈ zray which satisfies y ·e = 1.The horosphere Hp,q is the parabolic section of the restricted null cone,

Hp,q = y ∈ N0 | y ·e = 1,

see Figure 2. Thus Hp,q has dimension n = p+ q.The null cone N is determined by the condition y2 = 0, which taking

differencials gives

y ·dy = 0 ⇒ xc ·dy = 0 , (50)

where yray = xcray. SinceN0 is an (n+1)-dimensional surface, then(50) is a condition necessary and sufficient for a vector v to belong tothe tangent space to the restricted null cone T (N0) at the point y

v ∈ T (N0) ⇔ xc ·v = 0 . (51)

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26 J. Pozo and G. Sobczyk

It follows that the (n + 1)-pseudoscalar Iy of the tangent space to N0

at the point y can be defined by Iy = Ixc where I is the pseudoscalarof IRp+1,q+1. We have

xc ·v = 0 ⇔ 0 = I(xc ·v) = (Ixc)∧v = Iy∧v. (52)

RA

-e

ν

p,q

p,qe

p,q

e

σ

x

xh

x c

o

H

Figure 2. The restricted null cone and representants of the point x in affine spaceand on the horosphere.

4.2. the null cone

The mapping

c : IRp,q → N0 ⊂ IRp+1,q+1 , x 7→ c(x) ≡ xc (53)

is continuous and infinitely differentiable (indeed, its third differentialvanishes), and it is also an isometric embedding.

dxc = dx− x·dxe ⇒ (dxc)2 = (dx)2 (54)

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Geometric Algebra in Linear Algebra and Geometry 27

The mapping c(x) (53) constitutes a vectorial chart for the horosphere.The pseudoscalar Ixc of the tangent space to Hp,q at the point xc isgiven by

Ixc = IKx = I e∧xc. (55)

We can extend the mapping c(x) to give a scalar-vector chart forthe whole N0.

y : IRp,q × IR∗ → N0 , (x, t) 7→ y(x, t) ≡ tc(x) = txc (56)

4.3. h-twistors

Let us define the h-twistor to be a rotor Sx ∈ Spinp+1,q+1

Sx := 1 +12xe = exp (

12xe). (57)

Noting that SxSx† = 1, we define its angular velocity by

ΩS := 2Sx†dSx = dxe or equivalently ΩS(a) = ae ∀a ∈ IRp,q .(58)

Later, in section 4.7, we more carefully define an h-twistor to be anequivalence class of two “twistor” components from Gp,q, that havemany twistor-like properties.

The reason for these definitions are found in their properties. Thepoint xc is generated from 0c = e by

xc = SxeSx†, (59)

and the tangent space to the horosphere at the point xc is generatedfrom dx ∈ IRp,q by

dxc = dSx e Sx† + Sx e dSx

† = Sx(ΩS ·e)Sx† = SxdxSx† (60)

or, equivalently, in terms of the argument of the differential

dxc(a) = SxaSx† ∀a ∈ IRp,q.

It also keeps unchanged the “point at infinity” e

e = SxeSx†.

The motivation for the term “h-twistor” is that it generates bothpoints and tangent vectors on the horosphere from the correspondingobjects in IRp,q. We call the h-twistor (60) “non-rotational” because

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28 J. Pozo and G. Sobczyk

tangent vectors coincide with the differential of points. More generally,the h-twistor Tx := SxRx, with Rx ∈ Spin(IRp,q) generates

xc = TxeTx† = SxeSx

† and dxc(RxaRx†) = TxaTx†.

The angular velocity ΩT of the more general h-twistor Tx is easilycalculated

ΩT := 2Tx†dTx = Rx†ΩSRx + ΩR = Rx

†dxRxe+ ΩR. (61)

The analogy with Penrose twistors is, of course, not complete. We willhave more to say about this later.

4.4. Conformal Transformations and Isometries

In this subsection we show that every conformal transformation in IRp,q

corresponds to two isometries on the null cone N0 in IRp+1,q+1.

DEFINITION 1. A conformal transformation in IRp,q is any twicelydifferentiable mapping between two connected open subsets U and V ,

f : U −→ V, x 7−→ x′ = f(x)

such that the metric changes by only a conformal factor

(df(x))2 = λ(x)(dx)2, λ(x) 6= 0.

If p 6= q then λ(x) > 0. In the case p = q, there exists the posibility thatλ(x) < 0, when the conformal transformations belong to two disjointsubsets. We will only consider the case when λ(x) > 0.

Recall that N0 can be coordinized by the vector-scalar chart (56).Using the h-twistor (57), (59) and (60), we obtain the expressions

y = Sx te Sx† and dy = dtxc + tdxc = dtSxeSx

† + tSxdxSx†. (62)

It easily follows that

(dy)2 = t2(dxc)2 = t2(dx)2. (63)

DEFINITION 1.1. An isometry F on N0 is any twicely differentiablemapping between two connected open subsets U0 and V0 in the relativetopology of N0,

F : U0 −→ V0, y 7−→ y′ = F (y)

which satisfies (dF (y))2 = (dy)2 .

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Geometric Algebra in Linear Algebra and Geometry 29

Using the scalar-vector chart y(x, t) = txc, any mapping in N0 canbe expressed in the form

y′ = F (y) = t′x′c = φ(x, t)f(x, t)c

where t′ = φ(x, t) and x′c = f(x, t)c are defined implicitly by F . Using(63), we obtain the result that y′ = F (y) is an isometry if and only if

(dy′)2 = (dy)2 ⇔ t′2(dx′)2 = t2(dx)2 ⇔ (df(x, t))2 =

t2

φ(x, t)2(dx)2.

Since f(x, t), x ∈ IRp,q (non degenerate metric), and the right handside of this equation does not contain dt, it follows that f(x, t) = f(x) isindependent of t. It then follows that φ(x) := φ(x, t)/t is also indendentof t. Thus, we can express any isometry y′ = F (y) in the form y′ =tφ(x)f(x)c, where f(x)c ∈ N0 is the conformal representant of f(x) ∈IRp,q. This implies that y′ = F (y) is an isometry iff

y′ = tφ(x)f(x)c and (df(x))2 = (φ(x))−2(dx)2.

Therefore, f(x) is a conformal transformation with

λ(x) = φ(x)−2 > 0 ↔ φ(x) = ± 1√λ(x)

.

4.5. Isometries in N0

In this section we show that for any dimension greater than 2 anyisometry in N0 is the restriction of an isometry in IRp+1,q+1. Theinverse of the statement is obvious. From the definition of an isom-etry, (dF (x))2 = (dy)2. Since dF (y) and dy are vectors in IRp+1,q+1,dF (y) can be obtained as the result of applying a field of orthogonaltransformations to dy,

dF (y) = R(y)dyR(y)∗−1

(64)

expressed here through a field of versors R(y) ∈ Pinp+1,q+1 ≡ X =a1a2 · · · an ∈ Gp+1,q+1 | a2

i = ±1. Note that R(y)∗−1

= ±R(y)†,where R∗ and R† denote the main involution and the reversion respec-tively. Thus, the result that we must prove is that R(y) is constant, i.e.independent of the point y. This shall guarantee that F (y) is a globalrigid isometry.

The fact that the tangent space T (N0) has dimension n + 1 and ametrically degenerate null direction xc is sufficient to guarantee that the

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30 J. Pozo and G. Sobczyk

image of dF (y) defines a unique orthogonal transformation in IRp+1,q+1,which determines (up to a sign) the versor R(y) .

Previously, we found that any isometry F (y) = tφ(x)f(x)c in N0 islinear in the scalar coordinate t. Taking the exterior derivative, we get

dF (y) =dt

tF (y) + td (φ(x)f(x)c) =

dt

tF (y) + td

(φ(x)Sf(x)eSf(x)

†)

and using (64) and (62), we also have

R(y)dyR(y)∗−1

= dtt R(y)yR(y)∗

−1+ tR(y)SxdxSx†R(y)∗

−1.

It follows thatR(y)SxdxSx†R(y)∗−1

= d(φ(x)Sf(x)eSf(x)

†) and F (y) =

R(y)yR(y)∗−1

, so that R(y) is independent of t. We have now shownthat any isometry in N0 satisfies

dF (y) = R(x)dyR(x)∗−1

and F (y) = R(x)yR(x)∗−1

(65)

where R(x) ∈ Pinp+1,q+1 is solely a function of x ∈ IRp,q. It remainsto be shown that R(x) = R is also independent of x so that F (y) =RyR∗

−1 is a global orthogonal transformation in N0 ⊂ IRp+1,q+1.We now slightly generalize the definition of the h-twistor to apply

to the rotor Rx := R(x) ∈ Pinp+1,q+1. Letting Tx := RxSx, we canrewrite (65) in the form

dF (y) = Tx(t dx+ dt e)Tx∗−1 and F (y) = Tx te Tx

∗−1 (66)

Analogous to (58) and (61), we define the three bivector valued forms:

ΩR := 2R−1x dRx , Ω0 := Sx

†ΩRSx and ΩT := 2T−1x dTx (67)

From the definition of Tx, we obtain the relation

ΩT = Ω0 + ΩS .

In order to prove that Rx is constant, let us first impose the integra-bility condition that the second exterior differential ddF must vanish.Note that in the calculations below we are taking into account boththe antisymmetry of exterior forms as well as the non-commutativityof multivectors. Using (65), we find

0 = ddF = d(Rx dy Rx∗−1) = dRx dy Rx

∗−1 −Rx dy dRx∗−1

=12Rx (ΩR dy + dyΩR) Rx∗

−1 ⇒ ΩR ·dy = 0

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Geometric Algebra in Linear Algebra and Geometry 31

Using (62), this is equivalent to

ΩR ·dy = (Sx†ΩRSx)·(Sx†dySx) = Ω0 ·(t dx+ dt e) = 0 (68)

Since R(x) and Sx are independent of t, then Ω0 does not contain dt.Thus, equation (68) can be separated into two parts,

Ω0 ·(t dx+ dt e) = tΩ0 ·dx+ dtΩ0 ·e = 0 ⇒

Ω0 ·dx = 0Ω0 ·e = 0 (69)

From Ω0 ·e = 0, it follows that the bivector-valued form Ω0 can bewritten as

Ω0(x, a) = v(x, a)∧e+B(x, a)

where v(x, a) is a vector in IRp,q, and B(x, a) is a bivector in thegeometric algebra G2

p,q of IRp,q.Imposing the first equation in (69) we get

Ω0(a)·b− Ω0(b)·a = 0⇒v(a)·b− v(b)·a = 0B(a)·b−B(b)·a = 0⇒

⇒ B(a)·(b∧c) = B(b)·(a∧c)⇒ B(a) = 0 ∀a ∈ IRp,q

⇒ Ω0(a) = v(a)∧e. (70)

The second integrability condition is found by taking the exteriorderivative of ΩR = 2R−1

x dRx to find

dΩR = 2dRx−1dRx = 2dRx−1RxRx−1dRx = −1

2ΩRΩR. (71)

But (70) implies ΩRΩR = SxΩ0Ω0Sx† = 0, from which it follows that

dΩR = 0. Next, we write this as an equation in Ω0, getting:

0 = dΩR = d(SxΩ0Sx†) = Sx (dΩ0 + ΩS × Ω0)Sx†

⇔ dΩ0 + ΩS × Ω0 = 0.

With the help of (70) and (58), we now spit this equation into its threemultivector parts:

dΩ0 + ΩS ×Ω0 = dv e+ v∧dx+ v·dx e∧e = 0 ⇒

dv = 0v∧dx = 0v ·dx = 0

(72)

The bivector part

v∧dx = 0 ⇔ v(a)∧b = v(b)∧a

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32 J. Pozo and G. Sobczyk

differentiates drastically between the dimension d = 2, and for thedimensions d > 2. When d > 2, we can wedge this last expression withthe vector a ∈ IRp,q to infer

v(a)∧b∧a = 0 ∀ b ∈ IRp,q ⇒ v(a)∧a = 0

⇒ v(a) = ρa , ρ ∈ IR,

from which follows the desired result

ρa∧b = ρb∧a ⇒ ρ = 0 ⇒ v = 0 ⇒ Ω0 = 0.

Therefore R(x) is constant,

ΩR = 0⇒ dR(x) = 0⇒ R(y) = R = constant.

Thus, F (y) is a global orthogonal transformation in IRp+1,q+1,

F (y) = RyR∗−1

, R ∈ Pinp+1,q+1. (73)

Since the group of isometries in N0 is a double covering of the groupof Conformal transformations Conp,q in IRp,q, and the group Pinp+1,q+1

is a double covering of the group of orthogonal transformations O(p+1, q+1), it follows that Pinp+1,q+1 is a four-fold covering of Conp,q.

The case of d = 2 will be treated after introducing the matrixrepresentation of next section.

4.6. Matrix representation

The algebra Gp+1,q+1 is isomorphic to Gp,q⊗G1,1. This isomorphism canbe specified by means of the so called conformal split (Hestenes, 1991).Evidently, once this isomorphism of algebras is established, we canuse the matrix representation introduced in subsection 2.2 for SNB1,1,taking into account that the 2×2 matrices are defined over the moduleGp,q. This identification makes possible a very elegant treatment of theso-called Vahlen matrices (Lounesto, 1997; Maks, 1989; Cnops, 1996;Porteous, 1995).

The conformal split does not identify the algebra Gp,q appearing inthe isomorphism Gp,q⊗G1,1 directly with Gp,q := genIRp,q. Instead, theconformal split identifies Gp,q with a subalgebra of Gp+1,q+1 generatedby a subset of trivectors: Gp,q := genIRp,q u, where u = ση is theunit bivector orthogonal to IRp,q, as introduced in (36) and subsection4.1. This subalgebra has the property that it commutes with G1,1 =genσ, η so that

Gp+1,q+1 = Gp,q ⊗ G1,1.

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Geometric Algebra in Linear Algebra and Geometry 33

The multivectors belonging to the subalgebra Gp,q are characterizedby

A ∈ Gp,q ⇔ A = A+ + uA− , A+ ∈ G+p,q and A− ∈ G−p,q

where A+ and A− are, respectively, the even and odd multivectorsparts of the multivector A = A+ + A− ∈ Gp,q. Thus, we have a directcorrespondence between the multivector A ∈ Gp,q and the multivectorA ≡ A+ +A− ∈ Gp,q.

Recall that the idempotents u± = 12(1± u) of the algebra G1,1, first

defined in (20), satisfy the properties given in subsection (2.2):

u+ + u− = 1 , u+ − u− = u , u+u− = 0 = u−u+ , σu+ = u−σ ,

andu = e∧e , u+ =

12ee , u− =

12ee ,

ue = e = −eu , eu = e = −ue , σu+ = e , 2σu− = e .

The representation of G1,1, introduced in the subsection 2.2 usingthe spinor basis, enables us to write any multivector G ∈ Gp+1,q+1 =Gp,q ⊗ G1,1 in the form

G = ( 1 σ )u+

(A BC D

)(1σ

)where the entries of the 2× 2 matrix are in Gp,q. Noting that

u+A = u+(A+ + uA−) = u+(A+ +A−) = u+A ,

makes it possible to work directly with the proper subalgebra Gp,q,instead of having to deal with the extra complexity introduced by usingthe subalgebra Gp,q.

It follows that each multivector G ∈ Gp+1,q+1 can be written in theform

G = ( 1 σ )u+[G](

)= Au+ +Bu+σ + C∗u−σ +D∗u− (74)

where

[G] ≡(A BC D

)for A,B,C,D ∈ Gp,q.

The matrix [G] denotes the matrix corresponding to the multivector G,and as a consequence of the general argument given in (22), we havethe algebra isomorphism

[G1 +G2] = [G1] + [G2] and [G1G2] = [G1][G2],

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34 J. Pozo and G. Sobczyk

for all G1, G2 ∈ Gp+1,q+1 . This result is an example of the unusualfact that a matrix representation is sometimes possible even when themodule of components Gp,q does not commute with the subalgebra G1,1.Note, also, the relationships

u+[G] = [u+G] =(A B0 0

)and [G]u+ = [Gu+] =

(A 0C 0

).

The operation of reversion of multivectors translates into the fol-lowing transpose-like matrix operation:

if [G] =(A BC D

)then [G]† := [G†] =

(D BC A

)where A = A∗† is the Clifford conjugation.

4.7. h-twistors and Mobius transformations

As seen in section 4.3, the point xc ∈ Hp,q can be written in the form(59), xc = SxeSx

†. More generally, in the subsection 4.5, we saw thatany conformal transformation F (xc) must be of the form

sTxeTx† = F (xc) = φ(x)f(x)c = φ(x)Sf(x) e Sf(x)

† (75)

where s := TxTx = ±1.Using the matrix representation of the previous section, for a general

multivector G ∈ Gp+1,q+1, we find that

[GeG†] =(A BC D

)(0 01 0

)(D BC A

)=(BD

)(D B ) (76)

where

[e] =(

0 01 0

), [G] ≡

(A BC D

), [G]† =

(D BC A

).

The relationship (76) suggests defining the conformal h-twistor ofthe multivector G ∈ Gp+1,q+1 to be

[G]c :=(BD

),

which may also be identified with the multivector Gc := Ge = Bu+ +D∗e. The conjugate of the conformal h-twistor is then naturally definedby

[G]c† := (D B ) .

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Geometric Algebra in Linear Algebra and Geometry 35

conformal h-twistors give us a powerful tool for manipulating the con-formal representant and conformal transformations much more effi-ciently. For example, since xc is generated by the conformal h-twistor[Sx]c, it follows that

[xc] = [Sx]c[Sx]c† =

(x1

)( 1 −x ) =

(x −x2

1 −x

).

Two conformal h-twistors [G1]c and [G2]c will be said to be equiva-lent if they generate the same multivector, i.e., if

[G1]c[G1]c† = [G2]c[G2]c

† .

This is equivalent to the condition G1 eG1† = G2 eG2

†. Two conformalh-twistors [G1]c and [G2]c will be said to be projectively equivalent ifthey generate the same direction, i.e., if

[G1]c[G1]c† = ρ[G2]c[G2]c

† with ρ ∈ IR∗ .

This is equivalent to the condition G1 eG1†ray = G2 eG2

†ray.A sufficient condition for two spinor to be projectively equivalent is

the following:

If ∃H ∈ Gp,q such that HH ∈ IR and [G2]c = [G1]cH (77)

then [G2]c[G2]c† = HH [G1]c[G1]c

†.

Moreover, it is not difficult to show that if any component A,B,C or

D of the two conformal h-twistors(AB

)and

(CD

)is invertible, then

this condition is necessary and sufficient.We can now write the conformal transformation (75) in its spinorial

form[F (xc)] = φ(x)[Sf(x)]c[Sf(x)]c

† = s[Tx]c[Tx]c†,

from which it follows that [Tx]c and [Sf(x)]c are projectively equivalentspinors. Since the bottom component of

[Sf(x)]c =(f(x)

1

)is trivially invertible, the two spinors are equivalent by (77). Letting

[Tx]c =(MN

),

it follows that(MN

)=(f(x)

1

)H ⇒ H = N and f(x) = MN−1, (78)

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36 J. Pozo and G. Sobczyk

and also that φ(x) = sNN .The beautiful linear fractional expression for the conformal trans-

formation f(x),f(x) = (Ax+B)(Cx+D)−1 (79)

andφ(x) = s(Cx+D)(D − xC)

is a direct consequence of (78). Since Tx = RSx for the constant versor(73), R ∈ Pinp+1,q+1 , its spinorial form is given by

[Tx]c = [R][Sx]c =(A BC D

)(x1

)=(Ax+BCx+D

)=(MN

),

where

[R] =(A BC D

), for constants A,B,C,D ∈ Gp,q.

The linear fractional expression (79) extends to any dimension andsignature the well-known Mobius transformations in the complex plane.The components A,B,C,D of [R] are, of course, subject to the condi-tion that R ∈ Pinp+1,q+1.

Although more difficult to manipulate, our conformal h-twistors area generalization to any dimension and any signature of the familiar2-component spinors over the complex numbers, and the 4-componenttwistors. Penrose’s twistor theory (Penrose and MacCallum, 1972) hasbeen discussed in the framework of Clifford algebra by a number ofauthors, for example see (Ablamowicz and Salingaros, 1985), (Eds.Ablamowicz and Fauser, 2000, pp75-92). In the language of spinors,any null vector y ∈ N is the null pole of a conformal h-twistor, [y] =[G]c[G]c

†. Also, two h-twistors will define the same null pole if theydiffer only by a phase, [G2]c = [G1]cH, where HH = 1. To completethe analogy, note that each conformal h-twistor also defines a null flag,i.e. a null bivector, tangent to the null cone N . It easily follows fromthe expressions (59) and (60) that

xc dxc = Sx e dxSx† ⇒ [xc dxc] = [Sx]c dx [Sx]c

† .

Finally, any h-twistor differing only by a rotor H ∈ Spinp,q will givethe same null pole but with a different null flag, the null flag rotatedby the rotor H:

[Sx]cH dxH[Sx]c†

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Geometric Algebra in Linear Algebra and Geometry 37

4.8. The relative matrix representation

In the two preceding subsections, we have introduced and used a matrixrepresentation of Gp+1,q+1, based on the isomorphism Gp+1,q+1 ∼ Gp,q⊗G1,1. This matrix representation depends only upon the choice of a fixedspin basis in G1,1, but not on any basis of Gp,q. We can introduce analternative relative matrix representation relative to a choosen non nulldirection a ∈ IRp,q, by a slight modification of the former (74), namely,

G = ( 1 σa−1 )u+ [G]′(

1aσ

), (80)

so that

[G]′ =(

1 00 a

)[G](

1 00 a−1

).

Evidently, this relative representation has the disadvantage of de-pending on the direction a that is chosen. However, it has the importantadvantage that the parity of G ∈ Gp+1,q+1 is the same as the parity ofthe components A,B,C,D ∈ Gp,q, where

[G]′ =(A BC D

).

Moreover, it is more directly related to complex numbers and to the4-component twistors of (Penrose and MacCallum, 1972). This relativerepresentation will enable us to relate isometries on N0 for d = 2 withanalytic and antianalytic functions over the complex numbers IC or overthe dual numbers ID.

The vectorial representation of points is most directly related to thecomplex representation of points via the paravector representant of xrelative to a, defined by

zx := xa−1 ⇔ x = zxa . (81)

Whereas this definition is valid in any dimension, we only consider herethe dimension d = p + q = 2. The set of relative paravectors, in thiscase, is the even subalgebra:

zx | x ∈ IRp,q = G0p,q ⊕ G2

p,q = G+p,q for p+ q = 2 .

Depending on the signature, the square of the pseudoscalar I ∈ Gp,qcan be either negative (I2 = −1) or positive (I2 = 1). It follows thatthe algebra G+

p,q is isomorphic to either the complex numbers IC or tothe dual numbers

G+2,0 ' G

+0,2 ' IC and G+

1,1 ' ID .

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38 J. Pozo and G. Sobczyk

The two vectors a, Ia ∈ IRp,q constitutes an orthonormal basis. Rela-tive to this basis, the vector x and its paravector zx have the coordinateforms

x = x1a+ x2Ia and zx = x1 + x2I, (82)

where x1, x2 ∈ IR.For example, the relative matrix representation of the conformal

representant xc is

[xc]′ =(zx −zxzx1 −zx

)a.

The relative matrix representation of the reversion of (80) is

[G]′† := [G†]′ = a−1

(D −B−C A

)a.

Conformal h-twistors can also be defined for the relative matrix repre-sentation in the obvious way:

[G]′c :=(BD

)and [G]′c

† := (D −B ) ,

and satisfy[GeG†]′ = [G]′c [G]′c

† a.

4.9. Conformal transformations in dimension 2

Before restricting ourselves in subsection 4.5 to dimensions d > 2, wefound the expression (70)

Ω0 = v(a)e,

from which we derived the conditions (72). The expression for v canbe derived from the versor T ∈ Pinp+1,q+1 which generates (67) ΩT =Ω0 + ΩS .

Expressing the bivector (58) ΩS = dxe in terms of the relativeparavectors (81) zx = xa−1, we get

ΩS = dzxae .

From the definition (67) of ΩT , we find

dT =12TΩT =

12T (Ω0 + ΩS) =

12T (ve+ dzx ae)

where the parity of T ∈ Pinp+1,q+1 is even or odd. Let us define

G :=T , if T is evenaT , if T is odd (83)

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Geometric Algebra in Linear Algebra and Geometry 39

so that G is always even, and for which it is also true that dG = 12GΩT .

Using the relative matrix representation introduced earlier, we have

[G]′ ≡(A BC D

)and [ΩT ]′ =

(0 2dzx−av 0

)where A,B,C,D ∈ G+

p,q. Note that since the matrix representation (80)is defined in terms of constant vectors, the differential will commutewith the representation [dG]′ = d[G]′. It follows that(

dA dBdC dD

)=(A BC D

)(0 dzx−1

2av 0

).

We can split this matrix into two columns, getting(dAdC

)= −1

2av

(BD

)and

(dBdD

)=(AC

)dzx. (84)

Equation (84) implies that, considered as functions over G+p,q (iso-

morphic to IC or to ID), the two components B and D are analytic, sincetheir differentials are proportional to dzx. Therefore, the derivatives ofthese analytic functions are(

B′

D′

)=(AC

)where B′ :=

dB

dzx.

This implies, in turn, that the components A and C are also analytic(AC

)=(B′

D′

)and

(A′

C ′

)= −1

2av

dzx

(BD

). (85)

An immediate consequence of the above equations is that the 1-formav is proportional to dzx, so that Ω0 takes the form

av = g(zx)dzx ⇒ Ω0 = a−1e g(zx) dzx, (86)

where g(zx) is also an analytic function over G+p,q.

Taking into account the change of representation of the conformal

h-twistor [G]c to [G]′c , the formula (78) for the spinor [T ]′c =(MN

)becomes f(x) = MN−1a. Defining the function

f : G+p,q → G+

p,q , zx 7→ f(zx) := zf(x) = f(x)a−1,

we obtain f(zx) = MN−1.We must now consider the two cases when T in (83) is either odd

or even. If T is even then

T = G ⇒(MN

)=(BD

)⇒ f(zx) = BD−1.

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40 J. Pozo and G. Sobczyk

If T is odd then

T = a−1G ⇒(MN

)= a−1

(BD

)⇒ f(zx) = a−1BD−1a.

The function h(zx) defined by

h(zx) :=

f(zx) , if T is evena f(zx)a−1 = f(zx) , if T is odd

has the property that, regardless of whether T is even or odd,

h(zx) = BD−1 ⇒ B = h(zx)D . (87)

Since B and D are analytic, it follows that h(zx) is also analytic. In thecase that T is even, it generates the analytic transformation f(zx) =h(zx) in G+

p,q. On the other hand, in the case that T is odd, it generatesthe anti-analytic transformation f(zx) = h(zx).

Using (87) and (85), we can express [G]′ in terms of h(zx),

[G]′ =(A BC D

)=(

(hD)′ hDD′ D

). (88)

The fact that G is in Spinp,q can be used to find an explicit expressionfor D in terms of h(zx). Using that GG† = ±1,

[GG†]′ =(A BC D

)(D −B−C A

)=(AD −BC 0

0 AD −BC

),

it follows that det[G]′ ≡ AD−BC = ±1. From (85) and (87), it directlyfollows that

±1 = AD −BC = 2(B′D −BD′) = 2D2(BD−1)′ = D2h′

so that formally we have

D = ±(±h′)−12 = ± 1√

±h′(89)

which, in general, represents four solutions.For the complex case IC ' G+

2,0 ' G+0,2, where I2 = −1, the four

solutions of (89) are given as usual by

D =k√h′, where k = ±1,±I. (90)

The inverse and square roots of the dual number ±h′ ∈ ID ' G+1,1 in

(89), where I2 = 1, are not always well defined. The inverse of a dualnumber zx = x1 + x2I ∈ ID is given by

z−1x =

1zx

=z†x

x21 − x2

2

=x1 − x2I

x21 − x2

2

,

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Geometric Algebra in Linear Algebra and Geometry 41

so will only exist when x1 6= ±x2. It can be shown that the dual number±h′ (except in the degenerate case when h′h′† = 0) has exactly one ofthe four hyperbolic Euler forms (Sobczyk, 1995),

±h′ =±ρ exp(Iφ)±ρI exp(Iφ) ,

where ρ =√|h′h′†| and φ is the hyperbolic angle defined by ±h′. Only in

the case when the sign of ±h′ can be chosen such that ±h′ = ρ exp(Iφ),will ±h′ have four well-defined square roots in ID. For this case we have

D =k√±h′

=k√ρ

exp(−12Iφ), where k = ±1,±I. (91)

Once we have found D, we also have A, B and C (88)

B =kh√h′, A = k

(h′)2 − 12hh

′′

(h′)32

, C = −k h′′

2(h′)32

,

but it is not, in general, possible to solve for the transformation h(zx)which corresponds to a given Ω0 = a−1e g(zx) dzx. However, we can findg(zx) in terms of the function h(zx): From (85) and (86), we obtain thesecond order differential equation for the conformal h-twistor of G,(

B′′

D′′

)= −1

2g(zx)

(BD

). (92)

From (92) and (90) or (91), we have

g(zx) = −2D′′

D=h′′′

h′− 3

2

(h′′

h′

)2

.

It is recognized that g(zx) is the Schwarzian derivative of h(zx), whichvanishes whenever h(zx) is a Mobius transformation. There are manypossibilities for the further study of the Schwarzian derivative and itsgeneralizations (Kobayashi and Wada, 2000).

Acknowledgements

Jose Pozo acknowledges the support of the Spanish Ministry of Edu-cation (MEC), grant AP96-52209390, the project PB96-0384, and theCatalan Physics Society (IEC). Garret Sobczyk gratefully acknowledgesthe support of INIP of the Universidad de Las Americas-Puebla, andCIMAT-Guanajuato during his Sabbatical, Fall 1999.

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42 J. Pozo and G. Sobczyk

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