Geometric Algebra in Linear Algebra and Geometry

Home , Geometric algebra, Geometric Algebra (book), Matrix (mathematics), Multilinear algebra, Multivector, Outermorphism

Jos´eMar´ıaPozo 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 multilinear algebra, and in aﬃne, projective and conformal geometries. Our principal objective is to show how the rich algebraic tools of geometric algebra are fully compatible 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 geometry, geometric algebra, Grassmann algebra, horosphere, Lie algebra, linear algebra, Möbiustransformation, 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 2 J. Pozo and G. Sobczyk

aﬃne and projective geometries examples 4. Conformal Geometry − the horosphere the null cone h-twistors conformal transformations and isometries

isometries in 0 N matrix representation h-twistors and Mobius transformations the relative matrix representation conformal transformations in dimension 2

1. Introduction

Almost 125 years after the discovery of “geometric algebra” by William Kingdon Clifford in 1878, the discipline still languishes off the center- stage of mathematics. Whereas Clifford’s geometric algebra has gained currency among an increasing number scientists in different “special interest” groups, the authors of the present work contend that geometric algebra should be known by all mathematicians and other scientists for what it really is - the natural algebraic completion of the real number system to include the concept of direction. Whereas, evidently, most mathematicians and other scientists are either unfamiliar with or reject this point of view, we will try to prevail by showing that Clifford algebra really already has been universally recognized in the guise of linear algebra. Since linear algebra is fully compatible with Clifford algebra, it follows that in learning linear algebra, every scientist has really learned Clifford algebra but is generally unaware of this fact! What is lacking in the standard treatments of linear algebra is the recognition of the natural graded structure of linear algebra and, therefore, the geometric interpretation that goes along with the definition of geometric algebra. As has been often repeated by Hestenes and others, geometric algebra should 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 geometric algebra alongside the more traditional tools of linear algebra by taking full advantage of their fully compatible structures. There are many advantages to such an approach. First, everybody knows matrix algebra, but not everybody is aware that exactly the same algebraic rules apply to the multivectors in a geometric algebra. Because of this

gjfinPDF.tex; 6/02/2002; 14:45; p.2 Geometric Algebra in Linear Algebra and Geometry 3 fact, it is natural to consider matrices whose elements are taken from a geometric algebra. At the same time, by developing geometric algebra in such a way that any problem can be easily changed into an equivalent problem in matrix algebra, it becomes possible to utilize the powerful and extensive computer software that has been developed for working with matrices. Whereas CLICAL has proven itself to be a powerful computer aid in checking tedious Clifford algebra calculations, it lacks symbolic capabilities (Lounesto, 1994). Geometric algebra offers not only a comprehensive geometric interpretation but also a whole new set of algebraic tools for dealing with problems in linear algebra. We show that matrices, which are rectangular blocks of numbers, represent geometric numbers in a rather special spinor basis of a geometric algebra with neutral signature. This work consists of four main chapters. This introductory chapter lays down the rational for this article and gives a brief summary of its main ideas and content. Chapter 2 is primarily concerned with the development of the basic ideas of linear and multilinear algebra on an n-dimensional real vector space we call the null space, since we are assuming that all vectors in are null vectors (the square of each vector is zero). Taking all linear combinationsN of sums of products of vectors in generates the 2n-dimensional associative Grassmann algebra ( ). NThis stucture is sufficiently rich to efficiently develop many of theG basicN notions of linear algebra, such as the matrix of a linear operator and the theory of determinants and their properties. Recently, there has been much interest in the application of geometric 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 new computational tools for doing pseudeoeucliean and affine geometry using geometric algebra. Chapter 3 undertakes a systematic study of some of these models, and shows how the tools of geometric algebra make it possible to move freely between them, bringing a unification to the subject that is otherwise impossible. One of the key ideas is to define the meet and join operations on equivalence classes of blades of a geometric algebra which represent subspaces. Since a nonzero r- blade characterizes only the direction of a subspace, the magnitude of the blade is unimportant. Basic formulas for incidence relationships between points, lines, planes, and higher dimensional objects are compactly formulated. Examples of calculations are given in the affine plane which are just plain fun! Chapter 4 explores the deep relationships which exist between projective geometry and the conformal group. The conformal geometry of a pseudo-euclidean space can be linearized by considering the horosphere

gjfinPDF.tex; 6/02/2002; 14:45; p.3 4 J. Pozo and G. Sobczyk in a pseudo-Euclidean space of two dimensions higher. The introduction of 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 concept of an h-twistor greatly simpliﬁes calculations and is in many ways a generalization of the successful spinor/twistor formalisms to pseudoeuclidean spaces of arbitrary sig- natures. The utility of the h-twistor concept is amply demonstrated in a new derivation of the Schwarzian derivative (Davis, 1974, p46), (Nehari, 1952, p199).

2. Geometric Algebra and Matrices

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

e = ( e1 e2 en ) (1) { } ··· be a basis of . In this work we only consider real ( = IR) or complex ( = ICN) vector spaces although other fields couldK be chosen. By interpretingK each of the vectors in e to be the column vectors of the standard basis of the identity matrix{ } id(n) of the n n matrix algebra ( ) over the field , we are free to make the identification× e = idM(n).K We wish to emphasizeK that we are interpreting the basis { } vectors ei to be elements of the 1 n row matrix (1), and not the elements of a set. Thus, in what follows,× we are assumming and often will apply the rules of matrix multiplication when dealing with the (generalized) row vector of basis vectors e . Now let be the dual vector space of{ 1-forms} over the the field , and let e Nbe the dual basis of with respect to the basis e of .K If we now{ interpret} each of the vectorsN in e to be the row vectors{ } ofN the standard basis of the identity matrix id{(n}) of the n n matrix algebra ( ), we can again make the identification e =×id(n). Because we MwishK to be able to interpret the elements of e{ }as row vectors, we will always write the vectors in e in the column{ } vector form { } e1  e2  e =   (2) { }  ·   ·   en  We also assume that the column vector e obeys all the rules of matrix addition and multiplication of a n 1 column{ } vector. ×

gjfinPDF.tex; 6/02/2002; 14:45; p.4 Geometric Algebra in Linear Algebra and Geometry 5

In terms of these bases, any vector or point x can be written ∈ N x1  x2  n x = e x e = ( e1 e2 en ) = xiei (3) { } { } ··  ·  X   i=1  ·   xn  for xi IR, where ∈ x1  x2  x e =   { }  ·   ·   xn  are the column vector of components of the vector x with respect to the basis e . Since vectors{ } in are represented by column vectors, and vectors y by row vectors,N we deﬁne the operation of transpose of the vector x ∈byN

e1  e2  t t t x = ( e x e ) = x e e = ( x1 x2 . . . xn )   (4) { } { } { }{ }  ·   ·   en  In the case of the complex ﬁeld = , we have K C t x1  x2  x∗e = ( x1 x2 xn ) =   (5) { } ···  ·   ·   xn  The transpose and Hermitian transpose operations allows us to move between the reciprocal vector spaces and . Clearly the operation of Hermitian transpose reduces to the ordinaryN N transpose for real vectors. We now wish to weld together the structures of the matrix algebra ( ) and the geometric algebras generated by the vectors in the dual nullM K spaces and . Following (Doran, Hestenes, Sommen, Van Acker, 1993), we ﬁrstN considerN the Grassmann algebra ( ), generated by taking all linear combinations of sums and productsG N of the elements in the vector space = span e subject to the condition that for each x , x2 = xx =N 0. It follows{ } that ∈ N (x + y)2 = x2 + xy + yx + y2 = xy + yx = 0 (6)

gjfinPDF.tex; 6/02/2002; 14:45; p.5 6 J. Pozo and G. Sobczyk or xy = yx for all x, y in the null space . The geometric algebra ( ) generated− by a null space is calledN the Grassmann or exterior algebraG N for the null space . N N As follows from (6), the Grassmann exterior product a1a2 . . . ak of k vectors in is antisymmetric over the interchange of any two of its vectors; N a1 . . . ai . . . aj . . . ak = a1 . . . aj . . . ai . . . ak − so that the exterior product of null vectors is equivalent to the outer product of those vectors:

a1a2 . . . ak = a1 a2 ... ak. ∧ ∧ ∧ The 2n-dimensional standard basis SB e of ( ), is generated by taking all products of the vectors in the{ standard} G N basis e to get SB e = { } { } 1; e1, . . . , en; e12, . . . , e(n 1)n; ... ; e1 k, . . . , e(n k+1) n; ... ; e12 n = { − ··· − ··· ··· }

e , e , e ,..., en , (7) {{ 0} { 1} { 2} { }} n where ek := ( e1 k e(n k+1) n ) is the k -dimensional standard basis{ } of k-vectors··· ··· − ···

ej1j2 jk ej1 ej2 ejk ··· ≡ ··· n for the sets of indices 1 j1 < j2 < < jk n. In particular, it k ≤ ··· ≤ is assumed e = (1) and e = e . The unique component of en { 0} { 1} { } { } is the pseudoscalar or volume element I := e12 n. With respect to the standard basis SB(e) any multivector X (··· ) can be expressed in the matrix form ∈ G N X = SB e X SB (8) { } { } where X SB is the column vector of components { }

x 0 { }  x e  x { }  e2  X SB =  { }  { }    ·     ·  x en { } Just as we used the tranpose operation (4) to move from the the null space = span e to the dual null space , we can extend the deﬁnition ofN the transpose{ } to enable us to moveN from the Grassmann algebra ( ), to the Grassmann algebra ( ) of the reciprocal null G N G N

gjfinPDF.tex; 6/02/2002; 14:45; p.6 Geometric Algebra in Linear Algebra and Geometry 7 space . Since multivectors in ( ) are represented by column vectors, and multivectorsN Y ( ) byG N row vectors, we define the transpose Xt ( ) by ∈ G N ∈ G N t t t X = (SB e X SB ) = X SB SB e (9) { } { } { } { } t where X SB is the row vector of components { } t xt xt xt xt X SB = 0 e e en . { } { } { } { 2} ·· { } The Hermitian transpose is similarly defined when we are dealing with complex multivectors. The dual basis of multivectors SB e for ( ) are arranged in a column and are defined by { } G N

1  {e}  { } t  e2  SB e = (SB e ) =  { }  { } { }    ·   ·   en  { } where ek...1   e := · (10) { k}  ·   en...n k+1  − is the n -dimensional basis of dual k-vectors deﬁned by k

ej j ...j ej ej ej 1 2 k ≡ 1 2 ··· k n for the sets of indices n j1 > j2 > . . . > jk 1. k ≥ ≥ The dual space of the space , and more generally the dual Grassmann algebra N( ) of the GrassmannN algebra ( ), are deﬁned to satisfy the usualG propertiesN of the mathematical GdualN space. What (Doran, Hestenes, Sommen, Van Acker, 1993) observed was that these same properties can be faithfully expressed in a larger neutral geometric algebra n,n (a fomal deﬁnition is given below) containing both of these GrassmannG algebras as subalgebras, by replacing the duality conditions with corresponding reciprocal conditions. We accomplish all this by assuming the additional properties

2 2 ei = 0 = ei , eiej = ejei, eiej = ejei (for i = j), and eiej = ejei, − − 6 −(11)

gjfinPDF.tex; 6/02/2002; 14:45; p.7 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 algebra ( ) of the dual space becomes the natural reciprocal of the GrassmannG N algebra ( ). TheseN relations imply that the reciprocal k- vectors and k-formsG ofN Grassmann algebras ( ) and ( ) satisfy the reciprocal relations G N G N

e e = id( n n ) . { k}·{ k} k×k The neutral pseudoeuclidean space IRn,n is deﬁned as the linear space which contains both the null spaces and . Thus, N N IRn,n = = x + y x , y . N ⊕ N { | ∈ N ∈ N} 2n Likewise, the 2 –dimensional associative geometric algebra n,n is de- ﬁned to be the geometric algebra that contains both the GrassmannG algebras ( ) and ( ). We write G N G N

n,n = ( ) ( ) = gen e1, ..., en, e1, ..., en , (13) G G N ⊗ G N { } 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 super matrix algebra ( n,n). Recalling the basic geometric product of two vectors x, y, M G xy = x y + x y, (14) · ∧ we apply the same product to the of row and column basis vectors e and e , and simultaneously employ matrix multiplication, to get{ the} expressions{ }

e1 e1 e1 e2 ... e1 en ∧ ∧ ∧  e2 e1 e2 e2 ... e2 en  ∧ ∧ ∧ e e = e e + e e = id(n n)+ ......  { }{ } { }·{ } { }∧{ } ×    ......   en e1 en e2 ... en en  ∧ ∧ ∧ where id(n n) is the n n identity matrix, computed by taking all × × inner products ei ej between the basis vectors of e and e . Similarly, · { } { } n n n e e = e e + e e = ei ei + ei ei = n + ei ei, { }{ } { }·{ } { }∧{ } X · X ∧ X ∧ i=1 i=1 i=1

gjfinPDF.tex; 6/02/2002; 14:45; p.8 Geometric Algebra in Linear Algebra and Geometry 9 giving the useful formulas

n e e = n and e e = ei ei (15) { }·{ } { }∧{ } X ∧ i=1 Because of the metrical structure induced by the reciprocal relationships (12), we can express the components x e of the vector x (3) in the form { } ∈ N x1 e1 x ·  x2   e2 x  · x e =   =   = e x { }  ·   ·  { }·  ·   ·   xn   en x  · Similarly, the components of the reciprocal vector xt can be found from ∈ N

t t x e = ( x1 x2 . . . xn ) = x ( e1 e2 en ) (16) { } · ·· 2n We call n,n the universal geometric algebra of order 2 . When n is G countably inﬁnite, we call = , the universal geometric algebra. G G∞ ∞ The universal algebra contains all of the algebras n,n as proper G G subalgebras. In (Doran, Hestenes, Sommen, Van Acker, 1993), n,n is called the mother algebra. G

2.1. nondegenerate geometric algebras

The standard bases e and e of the reciprocal null spaces and , taken together, are said{ } to make{ } up a Witt basis of null vectorsN (Ablam-N owicz and Salingaros, 1985) of the neutral pseudoeuclidean space IRn,n. From the Witt basis, we can construct the standard orthonormal basis n,n of IR σ, η of n,n, { } G 1 1 σi = ei + ei ηi = ei ei (17) 2 − 2 for i = 1, 2, . . . , n. Using the deﬁning relationships (12) of the reciprocal frames e and e , we ﬁnd that these basis vectors satisfy { } { } 2 2 σi = 1 ηi = 1 ηiσj = σjηi i, j = 1, . . . , n − − ∀

σiσj = σjσi ηiηj = ηjηi i = j − − ∀ 6 The basis σ spans a real Euclidean vector space IRn and generates { } the geometric subalgebra n,0, whereas η spans an anti–Euclidean G { }

gjfinPDF.tex; 6/02/2002; 14:45; p.9 10 J. Pozo and G. Sobczyk

0,n space IR and generates the geometric subalgebra 0,n. The standard bases (7) of these geometric algebras naturally takeG the forms SB σ and SB η , { } { } so that a general multivector X n,0 can be written ∈ G X = SB σ XSB { } and similarly for an X n,0. We can now express the geometric ∈ G algebra n,n as the product of these geometric subalgebras G n,n = n,0 0,n = gen σ1, ..., σn, η1, ..., ηn , (18) G G ⊗ G { } 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 column and row vectors, respectively. When working in{ the} nondegenerate{ } geometric algebras n,n, n,0 or 0,n, we use the operation of reversal. G G G The reversal of any vector x n,n is deﬁned by x := x, and for the ∈ G † k-vector Ak = a1 a2 ... ak, ∧ ∧ ∧ k(k 1)/2 Ak† := ak ak 1 ... a1 = ( 1) − Ak. ∧ − ∧ ∧ −

2.2. Spinor basis

One nice application of the above formalism is that it allows us to simply express a natural isomorphism that exists between the neutral n n geometric algebra n,n, and the algebra of all real 2 2 matrices n G × n IR(2 ). To express this isomorphism, we first define 2 mutually commutingM idempotents 1 ui( ) = (1 σiηi) (19) ± 2 ± for i = 1, 2, . . . , n. We can now define 2n mutually annihiliating primitive idempotents for the algebra n,n, G u = u (signs ) (20) signs Y i i signs

th where signs is a particular sequence of n signs, and signsi is the i sign in the sequence. For example, ± n n u+++...+ = ui(+) and u ... = ui( ). Y −−− − Y − i=1 i=1 The primitive idempotents satisfy the following basic properties

gjfinPDF.tex; 6/02/2002; 14:45; p.10 Geometric Algebra in Linear Algebra and Geometry 11

2n usigns = 1 − Psigns

σiu+++...+ = u+...+ i+...+σi − − usign usign = δsign sign usign where δsign sign = 0 − 1 2 1 2 1 1 2 except when sign1 = sign2 for which δsign1 sign2 = 1. The above properties are easily veriﬁed. In contrast to the standard basis SB e of the neutral geometric { } n n algebra n,n, the spinor basis of n,n is deﬁned to be the 2 2 multivectorsG in the matrix G ×

t SNB(n, n) = SB σ u+++...+SB σ (21) { } { } The simplest example is the spinor basis for the geometric algebra 1 1,1. The 2 primitive idempotents for this geometric algebra are u = G1 (1 ση). Using (21), the spinor basis SNB(1, 1) is found to be ± 2 ±

t 1 u+ σu SB(σ) u+SB(σ) = u+ ( 1 σ ) = − σ 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 represents the × A corresponding element A n,n given by ∈ G t A = SB σ u+++...+ SB σ . { } A { } The matrix associated with the multivector A n,n is denoted by = [A]. ThisA association constitutes an algebra∈ isomorphism G , since [AA + B] = [A] + [B] and [AB] = [A][B]. Noting that

t n n u+ + SB σ SB σ u+ + = u+ + id(2 2 ), ··· { } { } ··· ··· × it easily follows that

t t AB = SB σ u+ + [A] SB σ SB σ u+ + [B] SB σ { } ··· { } { } ··· { } t = SB σ u+ + [A][B] SB σ (22) { } ··· { } We will use the spinor basis SNB(1, 1) for studying conformal transformation in section 4.

gjfinPDF.tex; 6/02/2002; 14:45; p.11 12 J. Pozo and G. Sobczyk

2.3. Symmetric and Hermitian Inner Products Until now we have only considered real geometric algebras and their corresponding real matrices. Any pseudoscalar of the geometric algebra n,n will always have a positive square, and will anticommute with theG vectors in IRn,n. If we insisted on dealing only with real geometric algebras, we might consider working in the geometric algebra n,n+1 where the pseudoscalar element i has the desired property that i2G= 1 and is in the center of the algebra (commutes with all multivectors).− A complex vector x + iy in n,n+1 consists of the real vector part x and a pseudovector or (2n)-bladeG part iy. Instead, we choose to directly complexify the geometric algebra n,n G to get the complex geometric algebra 2n(IC) (Sobczyk, 1996). Whereas G this algebra is isomorphic to n,n+1, it is somewhat easier to work with than the former. A complex Gvector z IC2n has the form z = x + iy where x, y IR2n. The imaginary unit∈i, where i2 = 1, is defined to ∈ − commute with all elements in the geometric algebra 2n(IC). 2n G Consider an orthonormal basis σ IC : σi σj = δij. The complexified null space (IC) and its{ } reciprocal ∈ null· space (IC) are the subspaces spanned byN the complex null vectors N 1 ej = (σj + iσn+j) and ej = σj iσn+j 2 − for j = 1, 2, . . . , n. This definition is consistent with (17) if we consider ηj = iσn+j. Thus a null vector x (IC) has the form x = e x e for ∈ N { } { } xi IC. ∈Previously we have defined the transposition (4). This operation can be extended to complex vectors in two different ways. The first way is a linear extension. We use the term transposition for the linear extension, so that the definition (4) is still valid when xi IC. The second extension is antilinear and is equivalent to Hermitian∈ conjugation: x x e . Both operations, Hermitian conjugation and ∗ ≡ ∗e { } transposition, take{ } us from the complex null space (IC) to the dual null space (IC), and if the components of x are allN real, both reduce N to the real transposition. Applied to the components x e , x∗e is the { } { } usual Hermitian transpose of the column vector x e , { } t x1  x2  x∗e = ( x1 x2 xn ) =   (23) { } ···  ·   ·   xn  We now define the symmetric inner product (x, y), and the Hermi- tian inner product x, y , on (IC). For all x, y (IC), the two prod- h i N ∈ N

gjfinPDF.tex; 6/02/2002; 14:45; p.12 Geometric Algebra in Linear Algebra and Geometry 13 ucts are deﬁned, respectively, by using transposition and Hermition conjugation:

t t (x, y) := x y = x e y e and x, y := x∗ y = x∗e y e (24) · { } { } h i · { } { } The Hermitian inner product will be used in the next subsection.

2.4. Linear Transformations

Let 0 and 0 be (n + n0)-dimensional reciprocal null spaces Nn+ ⊕n N,n+n N ⊕ N in IR 0 0 with the dual bases e e0 and e e0 . Let f : be a linear transformation{ } from ∪ { the} null{ space} ∪ { } into the N → N 0 N null space 0. In light of the previous section, we can consider the null spaces andN to be over the real or complex numbers. Let N N 0 Hom( , 0) = f : 0 f is a linear transformation N N { N → N | } denote the linear space of all homomorphisms from to 0, with the usual operation of addition of transformations. OfN course,N only when = is the operation of multiplication (composition) defined. N N 0 Given an operator f Hom( , 0), y0 = f(x) fx, the matrix of f with respect to the∈ bases eN andN e is defined≡ by F { } { 0} f e ( fe1 fen ) = ( e1 e0 ) = e0 . (25) { } ≡ ··· ··· n0 F { }F Of course, the matrix = (fij) is defined by its n n components F 0 × fij = ei f(ej) for i = 1, 2, . . . , n0 and j = 1, 2, . . . , n. It follows · ∈n0 C that f(ej) = i=1 ei0 fij. By dotting both sides of the above equation on the left byPe , we find the explicit expression { 0} = e0 e0 = e0 f e . F { }·{ }F { }· { } Equation (15) can be used to define the bivector F of the linear operator f. It is defined by

F = f e e0 { }∧{ } and satisfies the property that f x = F x for all x . The bivector of a linear operator makes possible a new· theory of∈ linearN operators, and is particularly useful in defining the general linear group as a Lie group 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 Hermitian transpose (23)) f ∗ : 0 of the mapping f : 0 is defined by the requirement thatN for→ all Nx and y , N → N ∈ N 0 ∈ N 0 x, f ∗(y0) = f(x), y0 ∗ e f ∗ e0 = [f e ]∗ e0 . h i h i ⇔ F ≡ { }· { } { } ·{ }

gjfinPDF.tex; 6/02/2002; 14:45; p.13 14 J. Pozo and G. Sobczyk

Likewise, we can deﬁne the transpose relative to the symmetric inner product

t t t t (x, f (y0)) = (f(x), y0) e f e0 = [f e ] e0 . ⇔ F ≡ { }· { } { } ·{ }

2.5. Outermorphism and generalized traces

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

f(x y z) f(x) f(y) f(z) x, y, . . . , z , ∧ ∧ · · · ∧ ≡ ∧ ∧ · · · ∧ ∀ ∈ N and where f(1) 1. Thus extended, f : ( ) ( ) is called ≡ G N → G N 0 the outermorphism of the linear transformation f : 0, since it preserves the structure of the outer product: N → N

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 directed areas, and more generally, directed k-vectors into directed k-vectors. A linear transformation from into itself is called an endomorphism. Let N

End( ) = f : f is a linear operator N { N → N | } denote the algebra of all endomorphisms on . The operations of addition and composition of linear operators isN well defined for endomorphisms. The determinant det f of the endomorphism f is defined to be the eigenvalue 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 f is defined by trf := f e e . Given the outermorphism of f we define the generalized traces{of}·{f by}

trif := f e e . { i}·{ i}

Particular cases are tr0f = f(1) 1 = 1 and tr1f = trf. The generalized · trace of degree n coincides with the determinant: trnf = f(I) I = det f. A second basis a of is related to the standard basis · e by the application of some{ endomorphism} N a { }

a = a e = e = ( e1 e2 en ) (26) { } { } { }A ··· A

gjfinPDF.tex; 6/02/2002; 14:45; p.14 Geometric Algebra in Linear Algebra and Geometry 15 where is called the matrix of transition from the basis e to the A n { } basis a . Taking the outer product i=1 a of the basis vectors a , we get{ } V { } { }

n a a1 a2 an = a(e1 e2 en) = det a I . (27) ^{ } ≡ ∧ ∧ · · · ∧ ∧ ∧ · · · ∧ i=1 We see from (27) that the determinant of the matrix of transition, det det a, between two bases cannot be zero. WeA ≡ can now easily construct a dual or reciprocal basis a for the basis a : { } { } i+1 (a1 ... i∗ ... an) I ai = ( 1) ∧ ∧ ∧ ∧ · (28) − det a where i∗ means that ai is omitted from the product. More compactly, using our matrix notation, a( e I) I a = { }· · . { } a(I) I · Checking, we ﬁnd that [ a( e I ) I ] a [ a( I e ) a e ] I a a = { }· · ·{ } = ·{ } ∧ { } · { }·{ } a(I) I a(I) I · · [ a(( I e ) e )] I a(I ( e e )) I = ·{ } ∧{ } · = { }·{ } · = e e = id a(I) I a(I) I { }·{ } · · We have actually found the inverse of the transition matrix , given by 1 = a e , (Eds. Bayro and Sobczyk, 2001, p.25). A A− { }·{ } 2.6. characteristic polynomial

The characteristic polynomial of f : is deﬁned by N → N ϕf (λ) = det(λ f) = (λ f)(I) I. − − · The well-known Caley-Hamilton theorem, which says that every linear operator satisﬁes its characteristic equation, is a consequence of the identity

f[x en 1 ] en 1 = (x en 1 ) en 1 det f = x det f (29) ∧{ − } ·{ − } ∧{ − } ·{ − } When the left side of this identity is expanded we get

n f[x e ] e = ( 1)i+1f e e f i(x) ∧{ n 1} ·{ n 1} X − { n i}·{ n i} − − i=1 − −

gjfinPDF.tex; 6/02/2002; 14:45; p.15 16 J. Pozo and G. Sobczyk

n i+1 i = ( 1) trn if f (x) . (30) X − − i=1 Expressed in terms of the generalized traces of f, the characteristic polynomial is

n i n i ϕf (λ) = (λ f)(e12 n) en 21 = ( 1) f ei ei λ − . − ··· · ··· X − { }·{ } i=0

Thus, from (29) and (30), we have ϕf (f) = 0, i.e. f satisﬁes its characteristic polynomial. The above equation (29) can also be used to derive a formula for the inverse of f. We get

1 (y f en 1 ) en 1 x = f (y) = ∧ { − } ·{ − }. − det f

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

r r ni mi ϕf (λ) = (λ λi) and ψf (λ) = (λ λi) Y − Y − i=1 i=1 where 1 mi ni n for i = 1, 2, . . . , r, and the roots λi are all distinct. ≤ ≤ ≤ The minimal polynomial uniquely determines, up to an ordering of the idempotents, the following spectral decomposition theorem of the linear operator f, (Sobczyk, 2001).

THEOREM 1. If f has the minimal polynomial ψ(λ), then a set of commuting mutually annihilating idempotents and corresponding nilpo- tents (pi, qi) i = 1, . . . , r can be found such that { | } r f = (λ + q )p , X i i i i=1 where rank(pi) = ni, and the index of nilpotency index(qi) = mi, for i = 1, 2, . . . , r. Furthermore, when mi = 1, qi = 0.

Clearly, the operator f is diagonalizable if and only if it has the spectral form r f = λ p . X i i i=1

gjfinPDF.tex; 6/02/2002; 14:45; p.16 Geometric Algebra in Linear Algebra and Geometry 17

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, we can immediately deﬁne a generalized inverse of the operator f by 1 q q inv i i mi 1 f = (pi + + (− ) − ) X λi − λi ··· λi λi=0 6 satisfying the conditions ff inv = f invf = p , (Rao and Mitra, λi=0 i 1971, pp.20). P 6

3. Geometric algebra and Non-Euclidean Geometry

Leonardo da Vinci (1452-1519) was one of the first to consider the problems of projective geometry. However, projective geometry was not formally developed until the work “Traitédes propriésprojectives des figure” of the French mathematician Poncelet (1788-1867), published in 1822. The extrordinary generality and simplicity of projective geometry led the English mathematician Cayley to exclaim: “Projective Geometry is all of geometry” (Young, 1930). n+1 Let IR be an (n + 1)-dimensional euclidean space and let n+1,0 be the corresponding geometric algebra. The directions or rays ofG non- zero vectors in IRn+1 are identified with the points of the n-dimensional projective plane Πn, (Hestenes and Ziegler, 1991). More precisely, we write n n+1 Π IR /IR∗ ≡ where IR∗ = IR 0 . We thus identify points, lines, planes, and higher dimensional k-planes− { } in Πn with 1, 2, 3, and (k + 1)-dimensional subspaces r of IRn+1, where k n. To effectively apply the tools of geometricS algebra, we need to≤ introduce the new basic operations of meet 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 easily deﬁned in terms of the intersection and direct sum of the subspaces which name the objects in Πn. On the other hand, each r-dimensional r n+1 subspace can be described by a non-zero r-blade Ar (IR ). We A ∈ G say that an r-blade Ar represents, or is a representant of an r-subspace r of IRn+1 if and only if A r n+1 = x IR x Ar = 0 . (31) A { ∈ | ∧ }

gjfinPDF.tex; 6/02/2002; 14:45; p.17 18 J. Pozo and G. Sobczyk

n+1 We denote the equivalence class of all nonzero r-blades Ar (IR ) which define the subspace r by ∈ G A Ar ray := tAr t IR, t = 0 . (32) { } { | ∈ 6 } Evidently, every r-blade in Ar ray is a representant of the subspace r. With these definitions, the{ problem} of finding the meet and join is reducedA to a problem in geometric algebra of finding the corresponding meet and join of the (r +1)- and (s+1)-blades in the geometric algebra (IRn+1) which represent these subspaces. G Let Ar, Bs and Ct be non-zero blades representing the three subspaces r, s and t, respectively. We say that A B C

DEFINITION 1. The t-blade Ct = Ar Bs is the meet of Ar and Bs ∩ if 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, and − ∧ ∧ Ac Bc = 0. ∧ 6

It is important to note that the t-blade Ct Ct ray is not unique and is deﬁned only up to a non-zero scalar∈ factor, { } which we choose at our own convenience. The existence of the t-blade Ct (and the corresponding complementary blades Ac and Bc) is an expression of the basic relationships that exists between subspaces.

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

Alternatively, since the join Ar Bs is defined only up to a non-zero ∪ scalar factor, we could equally well define D by D = Ac Bs. We use the symbols intersection and direct sum from set theory∧ to mark this unusual∩ state of affairs. The∪ problem of “meet” and “join” has thus been solved by finding the direct sum and intersection of linear subspaces and their (r + s t)-blade and t-blade representants. − Note that it is only in the special case when Ar Bs = 0 that the join 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 be ∪ ≡ ∪ used 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 deﬁnition of the meet and join of subspaces, the formula (33) holds only in IRn+1.

gjfinPDF.tex; 6/02/2002; 14:45; p.18 Geometric Algebra in Linear Algebra and Geometry 19

A slightly modified version of this formula will hold in any nondegenerate 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 a ∪ reciprocal (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 dimensional projective space Πn can be defined in an (n+1)-dimensional euclidean space IRn+1. There is a very close connection between affine and projective geometries. A projective space can be considered to be an affine space with idealized points at infinity (Young, 1930). Since all the formulas for meet and join remain valid in the pseudoeuclidean space IRp,q, subject only to (34), we will define the n = (p + q)- p,q 1 dimensional affine plane e(IR ) of the null vector e = 2 (σ + η) in the larger pseudoeuclideanA space IRp+1,q+1 = IRp,q IR1,1, where IR1,1 = span σ, η for σ2 = 1 = η2. Whereas, effectively,⊕ we are only extending the{ euclidean} space IR−p,q by the null vector e, it is advanta- geous to work in the geometric algebra p+1,q+1 of the non-degenerate pseudoeuclidean space IRp+1,q+1. G p,q p,q The affine plane e := e(IR ) is defined by A A p,q p,q p+1,q+1 e(IR ) = xh = x + e x IR IR , (35) A { | ∈ } ⊂ 1,1 p,q for the null vector e IR . The affine plane e(IR ) has the nice 2 2∈ p,q A property that xh = x for all xh e(IR ), thus preserving the metric structure of IRp,q. By employing∈ the Areciprocal null vector e = σ η with − p,q the property that e e = 1, we can restate definition (35) of e(IR ) in the form · A

p,q p+1,q+1 p+1,q+1 e(IR ) = y y IR , y e = 1 and y e = 0 IR A { | ∈ · · } ⊂ This form of the deﬁnition is interesting because it brings us closer to the deﬁnition of the n = (p + q)-dimensional projective plane . We summarize here the important properties of the reciprocal null 1 vectors e = 2 (σ + η) and e = σ η that will be needed later, and their relationship 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 points p,q of the aﬃne plane e(IR ), taken together with idealized points at A

gjfinPDF.tex; 6/02/2002; 14:45; p.19 20 J. Pozo and G. Sobczyk

p,q infinity. Each point xh e(IR ) is called a homogeneous representant of the corresponding point∈ A in Πn because it satisfies the property that xh e = 1. To bring these different viewpoints closer together, points in · p,q the affine plane e(IR ) will also be represented by rays in the space A rays p,q p+1,q+1 p+1,q+1 e (IR ) = y ray y IR , y e = 0, y e = 0 IR A {{ } | ∈ · · 6 } ⊂ (37) rays p,q The set of rays e (IR ) gives another definition of the affine n- A rays p,q plane, because each ray y ray e (IR ) determines the unique homogeneous point { } ∈ A y p,q yh = e(IR ). y e ∈ A · p,q Conversely, each point y e(IR ) determines a unique ray y ray in rays p,q ∈ A { }p,q e (IR ). Thus, the affine plane of homogeneous points e(IR ) is A rays p,q A equivalent to the affine plane of rays e (IR ). A h h h Suppose that we are given that we are given k-points a1 , a2 , . . . , ak p,q h p,q ∈ e(IR ) where each ai = ai+e for ai IR . Taking the outer product orA join of these points gives the projective∈ (k 1)-plane Ah Πn. Expanding the outer product gives − ∈

Ah = ah ah ... ah = ah (ah ah) ah ... ah 1 ∧ 2 ∧ ∧ k 1 ∧ 2 − 1 ∧ 3 ∧ ∧ k = ah (ah ah) (ah ah) ah ... ah = ... 1 ∧ 2 − 1 ∧ 3 − 2 ∧ 4 ∧ ∧ k h = a1 (a2 a1) (a3 a2) ... (ak ak 1), ∧ − ∧ − ∧ ∧ − − or h h h h A = a a ... a = a1 a2 ... ak+ 1 ∧ 2 ∧ ∧ k ∧ ∧ ∧ e (a2 a1) (a3 a2) ... (ak ak 1). (38) ∧ − ∧ − ∧ ∧ − − Whereas (38) represents a (k 1)-plane in Πn, it also belongs to the p,q − aﬃne (p, q)-plane e , and thus contains important metrical informa- tion. Dotting thisA equation with e, we ﬁnd that

h h h h e A = e (a1 a2 ... ak) = (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 = ah ah ... ah in the aﬃne (p, q)-plane is given by − 1 ∧ 2 ∧ ∧ k e Ah e (ah ah ... ah) · = · 1 ∧ 2 ∧ ∧ k (k 1)! (k 1)! − − (a2 a1) (a3 a2) ... (ak ak 1) = − ∧ − ∧ ∧ − − (k 1)! −

gjfinPDF.tex; 6/02/2002; 14:45; p.20 Geometric Algebra in Linear Algebra and Geometry 21

3.3. examples

p,q Many incidence relations can be expressed in the aﬃne plane e(IR ) which are also valid in the projective plane Πn, (Eds. Bayro andA Sobczyk, 2001, pp.263). A few examples are provided below. 2 Given 4 coplanar points ah, bh, ch, dh e(IR ). The join and meet ∈ A of 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 and join, we ﬁnd that∧ ∧

(ah bh) (ch dh) = det ah, bh, ch I = det a, b I (39) ∧ ∪ ∧ { } { } where I = σ1 σ2 e, and ∧ ∧ (ah bh) (ch dh) = det c d, b c ah + det c d, c a bh (40) ∧ ∩ ∧ { − − } { − − } Note that the meet (40) is not, in general, a homogeneous point. 2 Normalizing (40), we ﬁnd the homogeneous point ph e(IR ) ∈ A det c d, b c ah + det c d, c a bh p = { − − } { − − } h det c 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 noting that ph = αpah + (1 αp)bh = βpch + (1 βp)dh − − and solving to get αp = det bh, ch, dh / det bh ah, ch, dh . { 2 } { − } 2 Given the line ah bh e(IR ) and a third point dh e(IR ), as in ∧ ∈ A ∈ A Figure 1, the point fh on the line ah bh which is closest to the point dh ∧ is called the foot of the point dh on the line ah bh. Since fh ah bh = 0, ∧ ∧ ∧ it follows that fh = αf ah + (1 αf )bh and fh bh = αf ah bh. We can − ∧ ∧ solve this last equation for αf by dotting it with e, and invoking the auxilliary condition that (b f) (d f) = 0. We get − · − (a b).(d b) α = − − (41) f (a b)2 − 2 It should be carefully noted that ah bh = a b IR for any two 2 − − ∈ homogeneous points ah, bh . It follows that the foot fh on the line ∈ Ae ah bh is given by ∧ (b d) (b a)ah + (a d) (a b)bh f = − · − − · − . (42) h (a b)2 −

gjfinPDF.tex; 6/02/2002; 14:45; p.21 22 J. Pozo and G. Sobczyk

2 Saying that ah, bh, ch e are non-collinear points is equivalent ∈ A 2 to the condition ah bh ch = 0. If dh is any other point in , then ∧ ∧ 6 Ae 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 can ∧ ∧ easily solve for αd and βd, getting

det dh, bh, ch det dh, ch, ah αd = { } and βd = { } (43) det ah, bh, ch det ah, bh, ch { } { }

fh ph

Figure 1. Incidence relationships in the aﬃne plane.

2 Three non-collinear points ah, bh, ch determine a unique circle ∈ Ae with center rh = αrah + βrbh + (1 αr βr)ch. To ﬁnd the center, − − note that rh lies on the intersection of the perperdicular bisectors of the cords ah bh and ah ch, and therefore satisﬁes ∧ ∧ 1 1 rh = (ah + bh) + s(ch wh) = (ah + ch) + t(bh qh), (44) 2 − 2 − where

wh = fwah + (1 fw)bh, and qh = fqah + (1 fq)ch − − are the feet (42) of ch and bh along the lines ah bh and ah ch, respectively, for ∧ ∧ (a b (c b) (c a) (b a) f = − · − and f = − · − . w (a b)2) q (c a)2) − −

gjfinPDF.tex; 6/02/2002; 14:45; p.22 Geometric Algebra in Linear Algebra and Geometry 23

From (44), it follows that 1 1 (fws fqt)ah + [t + (1 fw)s ]bh + [ (1 fq)t s]ch 0, − − − 2 2 − − − ≡ which gives f f s = q and t = w . 2fq(1 fw) + 2fw 2fq(1 fw) + 2fw − − After simpliﬁcation, the center rh is found to be

[fq + fw 2fbfw]ah + fwbh + fbch rh = − . (45) 2[fq + fw fbfw] − Another theorem of interest is Simpson’s theorem for the circle. We have assembled all of the tools necessary for a proof of this venerable 2 theorem in the affine plane e(IR ), but we will not prove it here (Eds. Bayro and Sobczyk, 2001,A pp.39). Simpson’s theorem has also been 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 any real advantages to proving such theorems on the horosphere and not in the simpler affine plane. The issue at hand is how to best represent problems in distance geometry (Dress and Havel, 1993). Hestenes and Zigler have also given a proof of Desargues theorem in the projective plane Π2 (Hestenes and Ziegler, 1991), by using its representation in the euclidean space IR3. A proof of Desargues theorem rays p,q can also be given in the affine plane of rays e (IR ), (Eds. Bayro and Sobczyk, 2001, pp.37). The importance ofA such proofs is that even though geometric algebra is endowed with a metric, there is no reason why we cannot use the tools of euclidean space to give a proof of this metric independent result. Indeed, as has been emphasized by Hestenes and others (Barnabei, Brini, and Rota, 1985), all the results of linear algebra can be supplied with such a projective interpretation.

4. Conformal Geometry

The conformal geometry of a pseudo-Euclidean space can be linearized by considering the horosphere in a pseudo-Euclidean space of two dimensions higher. Because it is so easy to introduce extra orthogonal anticommuting vectors into a geometric algebra, without altering the structure of the geometric algebra in any other way, the framework of geometric algebra oﬀers a uniﬁcation to the subject that is impossible in other formalisms. The horosphere has recently attracted the attention

gjfinPDF.tex; 6/02/2002; 14:45; p.23 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 subsections 4.1 and 4.2. In subsection 4.3, the concept of an h-twistor is introduced which will greatly simplify computations. An h-twistor is a generalization of the Penrose twistor concept. In subsection 4.4, we give a simple proof, using only basic concepts from differential geometry developed in (Hestenes and Sobczyk, 1984), of an intriging result that relates conformal transformations in a pseudoeuclidean space to isometries in a pseudoeuclidean space of two higher dimensions. The original proof of this striking relationship was given by (Haantjes, 1937). In subsection 4.5, we show that for any dimension greater than two, that any isometry on the null cone can be extended to all of the pseudoeuclidean space. In subsections 4.6 and 4.7, we show the beautiful relationships that exists between Mobius transformations (linear fractional transformations) and their 2 2 matrix representation over a suitable geometric algebra. In a final subsection,× we explore how all of the formalism developed in the previous sections can be utilized in the characterization of conformal transformations of the pseudoeuclidean space IRp,q. We develop the theory in a novel way which suggests a non-trivial generalization of the theory of two-component spinors and 4-component twistors. Recall that a conformal transformation preserves angles between tangent vectors at each point (Lounesto and Springer, 1989; Porteous, 1995) . The utility of the h-twistor concept is amply demonstrated in a new derivation of the Schwarzian derivative. p,q p+1,q+1 We begin by defining the horosphere e in IR by moving up p,q p,q H from the affine plane e := e(IR ). A A 4.1. the horosphere

p+1,q+1 p+1,q+1 Let p+1,q+1 = gen(IR ) be the geometric algebra of IR , G p,q p,q and recall the definition (35) of the affine plane e := e(IR ) IRp+1,q+1. Any point y IRp+1,q+1 can be writtenA in theA form y =⊂ x + αe + βe, where x IR∈ p,q and α, β IR. p,q∈ ∈ The horosphere e is most directly defined by H p,q p,q 2 := xc = xh + βe xh and x = 0. (46) He { | ∈ Ae c } With the help of (36), the condition that 2 2 2 xc = (xh + βe) = x + 2β = 0 x2 p,q gives us immediately that β := 2 . Thus each point xc e has the form − ∈ H 2 2 xh x 1 xc = xh e = x + e e = xhexh. (47) − 2 − 2 2

gjfinPDF.tex; 6/02/2002; 14:45; p.24 Geometric Algebra in Linear Algebra and Geometry 25

The last equality on the right follows from

1 1 1 2 xhexh = [(xh e)xh + (xh e)xh] = xh x e. 2 2 · ∧ − 2 h p,q p,q Just as xh e is called the homogeneous representant of x IR , ∈ A ∈ the point xc is called the conformal representant of both the points p,q p,q p,q xh e and x IR . The set of all conformal representants := c(IR∈p,q A) is called∈ the horosphere . The horosphere p,q is a non-linearH p,q H p,q model of both the affine plane e and the pseudoeuclidean space IR . The horosphere n for the EuclideanA space IRn was first introduced by F.A. Wachter,H a student of Gauss, (Havel, 1995), and has been recently 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 xh by 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 ∈ := y IRp+1,q+1 y2 = 0 . N { ∈ | } The subset of containing all the representants y xc ray for any x IRp,q is definedN to be the set ∈ { } ∈ 0 = y y e = 0 = x IRp,q xc ray, N { ∈ N | · 6 } ∪ ∈ { } and is called the restricted null cone. The conformal representant of a null ray z ray is the representant y z ray which satisfies y e = 1. The horosphere{ } p,q is the parabolic section∈ { } of the restricted null· cone, H p,q = y 0 y e = 1 , H { ∈ N | · } see Figure 2. Thus p,q has dimension n = p + q. The null cone His determined by the condition y2 = 0, which taking differencials givesN

y dy = 0 xc dy = 0 , (50) · ⇒ · where y ray = xc ray. Since 0 is an (n+1)-dimensional surface, then (50) is{ a} condition{ } necessary andN suﬃcient for a vector v to belong to the tangent space to the restricted null cone ( 0) at the point y T N v ( 0) xc v = 0 . (51) ∈ T N ⇔ ·

gjfinPDF.tex; 6/02/2002; 14:45; p.25 26 J. Pozo and G. Sobczyk

It follows that the (n + 1)-pseudoscalar Iy of the tangent space to 0 N at the point y can be deﬁned by Iy = Ixc where I is the pseudoscalar of IRp+1,q+1. We have

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

p,q H -e x c

xh o Ap,q p,q e e x R

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

4.2. the null cone

The mapping

p,q p+1,q+1 c : IR , 0 IR , x c(x) xc (53) → N ⊂ 7→ ≡ is continuous and infinitely differentiable (indeed, its third differential vanishes), and it is also an isometric embedding.

2 2 dxc = dx x dxe (dxc) = (dx) (54) − · ⇒

gjfinPDF.tex; 6/02/2002; 14:45; p.26 Geometric Algebra in Linear Algebra and Geometry 27

The mapping c(x) (53) constitutes a vectorial chart for the horosphere. p,q The pseudoscalar Ixc of the tangent space to at the point xc is given by H Ix = IKx = I e xc. (55) c ∧ We can extend the mapping c(x) to give a scalar-vector chart for the whole 0. N p,q y : IR IR∗ 0 , (x, t) y(x, t) tc(x) = txc (56) × → N 7→ ≡

4.3. h-twistors

Let us deﬁne the h-twistor to be a rotor Sx Spin ∈ p+1,q+1 1 1 S := 1 + xe = exp ( xe). (57) x 2 2

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

p,q ΩS := 2Sx†dSx = dxe or equivalently ΩS(a) = ae a IR . ∀ ∈ (58) Later, in section 4.7, we more carefully deﬁne an h-twistor to be an equivalence class of two “twistor” components from p,q, that have many twistor-like properties. G The reason for these deﬁnitions are found in their properties. The point xc is generated from 0c = e by

xc = SxeSx†, (59) and the tangent space to the horosphere at the point xc is generated from 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

p,q dxc(a) = SxaSx† a IR . ∀ ∈ It also keeps unchanged the “point at inﬁnity” e

e = SxeSx†.

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

gjfinPDF.tex; 6/02/2002; 14:45; p.27 28 J. Pozo and G. Sobczyk tangent vectors coincide with the diﬀerential of points. More generally, p,q the h-twistor Tx := SxRx, with Rx Spin(IR ) generates ∈ xc = TxeTx† = SxeSx† and dxc(RxaRx†) = TxaTx†.

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

ΩT := 2Tx†dTx = Rx†ΩSRx + ΩR = Rx†dxRxe + ΩR. (61) The analogy with Penrose twistors is, of course, not complete. We will have more to say about this later.

4.4. Conformal Transformations and Isometries

In this subsection we show that every conformal transformation in IRp,q p+1,q+1 corresponds to two isometries on the null cone 0 in IR . N DEFINITION 1. A conformal transformation in IRp,q is any twicely diﬀerentiable mapping between two connected open subsets U and V ,

f : U V, x x0 = f(x) −→ 7−→ such that the metric changes by only a conformal factor

(df(x))2 = λ(x)(dx)2, λ(x) = 0. 6 If p = q then λ(x) > 0. In the case p = q, there exists the posibility that λ(x)6 < 0, when the conformal transformations belong to two disjoint subsets. We will only consider the case when λ(x) > 0. Recall that 0 can be coordinized by the vector-scalar chart (56). Using the h-twistorN (57), (59) and (60), we obtain the expressions

y = Sx te Sx† and dy = dtxc + tdxc = dtSxeSx† + tSxdxSx†. (62)

It easily follows that

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

DEFINITION 1.1. An isometry F on 0 is any twicely diﬀerentiable N mapping between two connected open subsets U0 and V0 in the relative topology of 0, N F : U0 V0, y y0 = F (y) −→ 7−→ which satisﬁes (dF (y))2 = (dy)2 .

gjfinPDF.tex; 6/02/2002; 14:45; p.28 Geometric Algebra in Linear Algebra and Geometry 29

Using the scalar-vector chart y(x, t) = txc, any mapping in 0 can be expressed in the form N

y0 = F (y) = t0xc0 = φ(x, t)f(x, t)c where t0 = φ(x, t) and xc0 = f(x, t)c are deﬁned implicitly by F . Using (63), we obtain the result that y0 = F (y) is an isometry if and only if

2 2 2 2 2 2 2 2 t 2 (dy0) = (dy) t0 (dx0) = t (dx) (df(x, t)) = (dx) . ⇔ ⇔ φ(x, t)2 Since f(x, t), x IRp,q (non degenerate metric), and the right hand side of this equation∈ does not contain dt, it follows that f(x, t) = f(x) is independent of t. It then follows that φ(x) := φ(x, t)/t is also indendent of t. Thus, we can express any isometry y0 = F (y) in the form y0 = tφ(x)f(x)c, where f(x)c 0 is the conformal representant of f(x) p,q ∈ N ∈ IR . This implies that y0 = F (y) is an isometry iﬀ

2 2 2 y0 = tφ(x)f(x)c and (df(x)) = (φ(x))− (dx) .

Therefore, f(x) is a conformal transformation with

2 1 λ(x) = φ(x)− > 0 φ(x) = . ↔ ± λ(x) p

4.5. Isometries in 0 N In this section we show that for any dimension greater than 2 any p+1,q+1 isometry in 0 is the restriction of an isometry in IR . The inverse of theN statement is obvious. From the deﬁnition of an isometry, (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 orthogonal transformations to dy,

1 dF (y) = R(y)dyR(y)∗− (64) expressed here through a ﬁeld of versors R(y) P inp+1,q+1 X = 2 ∈ 1 ≡ { a1a2 an p+1,q+1 a = 1 . Note that R(y)∗− = R(y)†, ··· ∈ G | i ± } ± where R∗ and R† denote the main involution and the reversion respectively. 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 global rigid isometry. The fact that the tangent space ( 0) has dimension n + 1 and a T N metrically degenerate null direction xc is suﬃcient to guarantee that the

gjfinPDF.tex; 6/02/2002; 14:45; p.29 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 0 is linear in the scalar coordinate t. Taking the exterior derivative, weN get dt dt dF (y) = F (y) + td (φ(x)f(x) ) = F (y) + td φ(x)S eS t c t f(x) f(x)† and using (64) and (62), we also have

1 1 1 − dt − − . R(y)dyR(y)∗ = t R(y)yR(y)∗ + tR(y)SxdxSx†R(y)∗

1 It follows that R(y)SxdxSx†R(y)∗− = d φ(x)Sf(x)eSf(x)† and F (y) = 1 R(y)yR(y)∗− , so that R(y) is independent of t. We have now shown that any isometry in 0 satisﬁes N 1 1 dF (y) = R(x)dyR(x)∗− and F (y) = R(x)yR(x)∗− (65)

p,q where R(x) P inp+1,q+1 is solely a function of x IR . It remains to be shown∈ that R(x) = R is also independent of∈x so that F (y) = 1 p+1,q+1 RyR∗− is a global orthogonal transformation in 0 IR . We now slightly generalize the deﬁnition of theN h-twistor⊂ to apply to the rotor Rx := R(x) P inp+1,q+1. Letting Tx := RxSx, we can rewrite (65) in the form ∈

1 1 dF (y) = Tx(t dx + dt e)Tx∗− and F (y) = Tx te Tx∗− (66)

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

1 1 ΩR := 2Rx− dRx , Ω0 := Sx†ΩRSx and ΩT := 2Tx− 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 integrability condition that the second exterior differential ddF must vanish. Note that in the calculations below we are taking into account both the antisymmetry of exterior forms as well as the non-commutativity of multivectors. Using (65), we find

1 1 1 0 = ddF = d(Rx dy Rx∗− ) = dRx dy Rx∗− Rx dy dRx∗− − 1 1 = Rx (ΩR dy + dy ΩR) Rx∗− ΩR dy = 0 2 ⇒ ·

gjfinPDF.tex; 6/02/2002; 14:45; p.30 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 dx = 0 Ω0 (t dx + dt e) = t Ω0 dx + dt Ω0 e = 0 · (69) · · · ⇒ Ω0 e = 0 · From Ω0 e = 0, it follows that the bivector-valued form Ω0 can be written 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 the 2 p,q geometric algebra p,q of IR . Imposing the ﬁrstG equation in (69) we get

v(a) b v(b) a = 0 Ω0(a) b Ω0(b) a = 0 · − · · − · ⇒ B(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 exterior 1 derivative of ΩR = 2Rx− dRx to ﬁnd

1 1 1 1 dΩR = 2dRx− dRx = 2dRx− RxRx− dRx = ΩRΩR. (71) −2

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 three multivector parts:

dv = 0  dΩ0 + ΩS Ω0 = dv e + v dx + v dx e e = 0  v dx = 0 (72) × ∧ · ∧ ⇒ v∧dx = 0  · The bivector part

v dx = 0 v(a) b = v(b) a ∧ ⇔ ∧ ∧

gjfinPDF.tex; 6/02/2002; 14:45; p.31 32 J. Pozo and G. Sobczyk diﬀerentiates drastically between the dimension d = 2, and for the dimensions d > 2. When d > 2, we can wedge this last expression with the 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,

1 F (y) = RyR∗− ,R P inp+1,q+1. (73) ∈

Since the group of isometries in 0 is a double covering of the group N p,q of Conformal transformations Conp,q in IR , and the group P inp+1,q+1 is a double covering of the group of orthogonal transformations O(p+ 1, q+1), it follows that P inp+1,q+1 is a four-fold covering of Conp,q. The case of d = 2 will be treated after introducing the matrix representation of next section.

4.6. Matrix representation

The algebra p+1,q+1 is isomorphic to p,q 1,1. This isomorphism can be specified byG means of the so called Gconformal⊗G split (Hestenes, 1991). Evidently, once this isomorphism of algebras is established, we can use the matrix representation introduced in subsection 2.2 for SNB1,1, taking into account that the 2 2 matrices are defined over the module × p,q. This identification makes possible a very elegant treatment of the Gso-called Vahlen matrices (Lounesto, 1997; Maks, 1989; Cnops, 1996; Porteous, 1995). The conformal split does not identify the algebra p,q appearing in Gp,q the isomorphism p,q 1,1 directly with p,q := gen IR . Instead, the G ⊗G G { } conformal split identifies p,q with a subalgebra of p+1,q+1 generated G p,q G by a subset of trivectors: Gp,q := gen IR u , where u = ση is the unit bivector orthogonal to IRp,q, as introduced{ } in (36) and subsection 4.1. This subalgebra has the property that it commutes with 1,1 = gen σ, η so that G { } p+1,q+1 = Gp,q 1,1. G ⊗ G

gjfinPDF.tex; 6/02/2002; 14:45; p.32 Geometric Algebra in Linear Algebra and Geometry 33

The multivectors belonging to the subalgebra Gp,q are characterized by

+ + + A Gp,q A = A + uA− ,A and A− − ∈ ⇔ ∈ Gp,q ∈ Gp,q + where A and A− are, respectively, the even and odd multivectors + parts of the multivector A = A + A p,q. Thus, we have a direct − ∈ G correspondence between the multivector A Gp,q and the multivector + ∈ A A + A− p,q. ≡ ∈ G 1 Recall that the idempotents u = 2 (1 u) of the algebra 1,1, ﬁrst deﬁned in (20), satisfy the properties± given± in subsection (2.2):G

u+ + u = 1 , u+ u = u , u+u = 0 = u u+ , σu+ = u σ , − − − − − − and 1 1 u = e e , u+ = ee , u = ee , ∧ 2 − 2 ue = e = eu , eu = e = ue , σu+ = e , 2σu = e . − − − The representation of 1,1, introduced in the subsection 2.2 using G the spinor basis, enables us to write any multivector G p+1,q+1 = ∈ G Gp,q 1,1 in the form ⊗ G AB 1 G = ( 1 σ ) u + CD σ 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 p,q, instead of having to deal with the extra complexity introduced by usingG the subalgebra Gp,q. It follows that each multivector G p+1,q+1 can be written in the form ∈ G 1 G = ( 1 σ ) u+[G] = Au+ + Bu+σ + C∗u σ + D∗u (74) σ − − where AB [G] for A, B, C, D p,q. ≡ CD ∈ G The matrix [G] denotes the matrix corresponding to the multivector G, and as a consequence of the general argument given in (22), we have the algebra isomorphism

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

gjfinPDF.tex; 6/02/2002; 14:45; p.33 34 J. Pozo and G. Sobczyk for all G1,G2 p+1,q+1 . This result is an example of the unusual fact that a matrix∈ G representation is sometimes possible even when the module of components p,q does not commute with the subalgebra 1,1. Note, also, the relationshipsG G AB A 0 u [G] = [u G] = and [G]u = [Gu ] = . + + 0 0 + + C 0 The operation of reversion of multivectors translates into the following transpose-like matrix operation: AB D B if [G] = then [G] := [G ] = CD † † C A where A = A∗† is the Cliﬀord conjugation.

4.7. h-twistors and Mobius transformations

As seen in section 4.3, the point xc p,q can be written in the form ∈ H (59), xc = SxeSx†. More generally, in the subsection 4.5, we saw that any 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 p+1,q+1, we ﬁnd that ∈ G AB 0 0 D B [GeG ] = † CD 1 0 C A

B = ( D B ) (76) D where 0 0 AB D B [e] = , [G] , [G]† = . 1 0 ≡ CD C A The relationship (76) suggests defining the conformal h-twistor of the multivector G p+1,q+1 to be ∈ G B [G] := , c D which may also be identified with the multivector Gc := Ge = Bu+ + D∗e. The conjugate of the conformal h-twistor is then naturally defined by [G]c† := ( D B ) .

gjfinPDF.tex; 6/02/2002; 14:45; p.34 Geometric Algebra in Linear Algebra and Geometry 35 conformal h-twistors give us a powerful tool for manipulating the conformal representant and conformal transformations much more eﬃ- ciently. For example, since xc is generated by the conformal h-twistor [Sx]c, it follows that x x x2 [xc] = [Sx] [Sx]† = ( 1 x ) = − . c c 1 − 1 x − Two conformal h-twistors [G1]c and [G2]c will be said to be equivalent if they generate the same multivector, i.e., if

[G1]c[G1]c† = [G2]c[G2]c† .

This is equivalent to the condition G1 e G1† = G2 e G2†. Two conformal h-twistors [G1]c and [G2]c will be said to be projectively equivalent if they generate the same direction, i.e., if

[G1] [G1]† = ρ[G2] [G2]† with ρ IR∗ . c c c c ∈ This is equivalent to the condition G1 e G1† ray = G2 e G2† ray. A suﬃcient condition for two spinor{ to be} projectively{ equivalent} is the following:

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

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 A C D of the two conformal h-twistors and is invertible, then B D this condition is necessary and suﬃcient. We can now write the conformal transformation (75) in its spinorial form [F (x )] = φ(x)[S ] [S ] = s[T ] [T ] , c f(x) c f(x) c† x c x c† from which it follows that [T ] and [S ] are projectively equivalent x c f(x) c spinors. Since the bottom component of f(x) [S ] = f(x) c 1 is trivially invertible, the two spinors are equivalent by (77). Letting M [T ] = , x c N it follows that

M f(x) 1 = H H = N and f(x) = MN − , (78) N 1 ⇒

gjfinPDF.tex; 6/02/2002; 14:45; p.35 36 J. Pozo and G. Sobczyk and also that φ(x) = sNN. The beautiful linear fractional expression for the conformal transformation f(x), 1 f(x) = (Ax + B)(Cx + D)− (79) and φ(x) = s(Cx + D)(D xC) − is a direct consequence of (78). Since Tx = RSx for the constant versor (73), R P inp+1,q+1 , its spinorial form is given by ∈ AB x Ax + B M [T ] = [R][S ] = = = , x c x c CD 1 Cx + D N where AB [R] = , for constants A, B, C, D p,q. CD ∈ G The linear fractional expression (79) extends to any dimension and signature the well-known Mobius transformations in the complex plane. The components A, B, C, D of [R] are, of course, subject to the condition that R P inp+1,q+1. Although∈ more difficult to manipulate, our conformal h-twistors are a generalization to any dimension and any signature of the familiar 2-component spinors over the complex numbers, and the 4-component twistors. Penrose’s twistor theory (Penrose and MacCallum, 1972) has been discussed in the framework of Clifford algebra by a number of authors, for example see (Ablamowicz and Salingaros, 1985), (Eds. Ablamowicz and Fauser, 2000, pp75-92). In the language of spinors, any null vector y is the null pole of a conformal h-twistor, [y] = ∈ N [G]c[G]c†. Also, two h-twistors will define the same null pole if they differ only by a phase,[G2]c = [G1]cH, where HH = 1. To complete the analogy, note that each conformal h-twistor also defines a null flag, i.e. a null bivector, tangent to the null cone . It easily follows from the expressions (59) and (60) that N

xc dxc = Sx e dx Sx† [xc dxc] = [Sx] dx [Sx]† . ⇒ c c Finally, any h-twistor differing only by a rotor H Spinp,q will give the same null pole but with a different null flag, the∈ null flag rotated by the rotor H: [Sx]c H dx H[Sx]c†

gjfinPDF.tex; 6/02/2002; 14:45; p.36 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 matrix representation of p+1,q+1, based on the isomorphism p+1,q+1 p,q G G ∼ G ⊗ 1,1. This matrix representation depends only upon the choice of a fixed G spin basis in 1,1, but not on any basis of p,q. We can introduce an alternative relativeG matrix representation relativeG to a choosen non null direction a IRp,q, by a slight modification of the former (74), namely, ∈ 1 G = ( 1 σa 1 ) u [G] , (80) − + 0 aσ so that 1 0 1 0 [G]0 = [G] 1 . 0 a 0 a− Evidently, this relative representation has the disadvantage of depending on the direction a that is chosen. However, it has the important advantage that the parity of G p+1,q+1 is the same as the parity of ∈ G the components A, B, C, D p,q, where ∈ G AB [G] = . 0 CD Moreover, it is more directly related to complex numbers and to the 4-component twistors of (Penrose and MacCallum, 1972). This relative representation will enable us to relate isometries on N0 for d = 2 with analytic and antianalytic functions over the complex numbers IC or over the dual numbers ID. The vectorial representation of points is most directly related to the complex representation of points via the paravector representant of x relative to a, defined by

1 zx := xa− x = zxa . (81) ⇔ Whereas this deﬁnition is valid in any dimension, we only consider here the dimension d = p + q = 2. The set of relative paravectors, in this case, is the even subalgebra:

p,q 0 2 + zx x IR = = for p + q = 2 . { | ∈ } Gp,q ⊕ Gp,q Gp,q Depending on the signature, the square of the pseudoscalar I p,q can be either negative (I2 = 1) or positive (I2 = 1). It follows∈ that G + − the algebra p,q is isomorphic to either the complex numbers IC or to the dual numbersG

+ + IC and + ID . G2,0 'G0,2 ' G1,1 '

gjfinPDF.tex; 6/02/2002; 14:45; p.37 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 coordinate forms 1 2 1 2 x = x a + x Ia and zx = x + x I, (82) where x1, x2 IR. For example,∈ the relative matrix representation of the conformal representant xc is zx zxzx [xc]0 = − a. 1 zx − The relative matrix representation of the reversion of (80) is

1 D B [G] † := [G ] = a a. 0 † 0 − C −A − Conformal h-twistors can also be deﬁned for the relative matrix representation in the obvious way: B [G]0 := and [G]0 † := ( D B ) , c D c − and satisfy [GeG†]0 = [G]c0 [G]c0 † a.

4.9. Conformal transformations in dimension 2

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

Ω0 = v(a)e, from which we derived the conditions (72). The expression for v can be derived from the versor T P inp+1,q+1 which generates (67) ΩT = ∈ Ω0 + ΩS . Expressing the bivector (58) ΩS = dxe in terms of the relative 1 paravectors (81) zx = xa− , we get

ΩS = dzxae .

From the definition (67) of ΩT , we find 1 1 1 dT = T Ω = T (Ω + Ω ) = T (ve + dz ae) 2 T 2 0 S 2 x where the parity of T P inp+1,q+1 is even or odd. Let us define ∈ T , if T is even G := (83) aT , if T is odd

gjfinPDF.tex; 6/02/2002; 14:45; p.38 Geometric Algebra in Linear Algebra and Geometry 39

1 so that G is always even, and for which it is also true that dG = 2 GΩT . Using the relative matrix representation introduced earlier, we have

AB 0 2dzx [G]0 and [ΩT ]0 = ≡ CD av 0 − + where A, B, C, D p,q. Note that since the matrix representation (80) is defined in terms∈ G of constant vectors, the differential will commute with the representation [dG]0 = d[G]0. It follows that dA dB AB 0 dz = x . dC dD CD 1 av 0 − 2 We can split this matrix into two columns, getting dA 1 B dB A = av and = dzx. (84) dC −2 D dD C + Equation (84) implies that, considered as functions over p,q (isomorphic to IC or to ID), the two components B and D are analytic,G since their differentials are proportional to dzx. Therefore, the derivatives of these analytic functions are

B0 A dB = where B0 := . D0 C dzx This implies, in turn, that the components A and C are also analytic A B A 1 av B = 0 and 0 = . (85) C D0 C0 −2 dzx D An immediate consequence of the above equations is that the 1-form av is proportional to dzx, so that Ω0 takes the form 1 av = g(zx)dzx Ω0 = a− e g(zx) dzx, (86) ⇒ + where g(zx) is also an analytic function over p,q. Taking into account the change of representationG of the conformal M h-twistor [G] to [G] , the formula (78) for the spinor [T ] = c c0 c0 N 1 becomes f(x) = MN − a. Deﬁning the function + + 1 f : , zx f(zx) := z = f(x)a− , Gp,q → Gp,q 7→ f(x) 1 we obtain f(zx) = MN − . We must now consider the two cases when T in (83) is either odd or even. If T is even then

M B 1 T = G = f(zx) = BD− . ⇒ N D ⇒

gjfinPDF.tex; 6/02/2002; 14:45; p.39 40 J. Pozo and G. Sobczyk

If T is odd then

1 M 1 B 1 1 T = a− G = a− f(zx) = a− BD− a. ⇒ N D ⇒

The function h(zx) deﬁned by

f(zx) , if T is even h(zx) := 1 a f(zx)a− = f(zx) , if T is odd has the property that, regardless of whether T is even or odd, 1 h(zx) = BD− B = h(zx)D. (87) ⇒ Since B and D are analytic, it follows that h(zx) is also analytic. In the case that T is even, it generates the analytic transformation f(zx) = + h(zx) in . On the other hand, in the case that T is odd, it generates Gp,q the anti-analytic transformation f(zx) = h(zx). Using (87) and (85), we can express [G]0 in terms of h(zx),

AB (hD)0 hD [G]0 = = . (88) CD D0 D

The fact that G is in Spinp,q can be used to ﬁnd an explicit expression for D in terms of h(zx). Using that GG = 1, † ± AB D B AD BC 0 [GG ] = = , † 0 CD CA− −0 AD BC − − it follows that det[G]0 AD BC = 1. From (85) and (87), it directly follows that ≡ − ± 2 1 2 1 = AD BC = 2(B0D BD0) = 2D (BD− )0 = D h0 ± − − so that formally we have

1 1 D = ( h0)− 2 = (89) ± ± ±√ h ± 0 which, in general, represents four solutions. + + 2 For the complex case IC 2,0 0,2, where I = 1, the four solutions of (89) are given as'G usual by'G − k D = , where k = 1, I. (90) √h0 ± ± The inverse and square roots of the dual number h ID + in ± 0 ∈ 'G1,1 (89), where I2 = 1, are not always well deﬁned. The inverse of a dual number zx = x1 + x2I ID is given by ∈ 1 1 zx† x1 x2I zx− = = 2 2 = 2− 2 , zx x x x x 1 − 2 1 − 2

gjfinPDF.tex; 6/02/2002; 14:45; p.40 Geometric Algebra in Linear Algebra and Geometry 41 so will only exist when x1 = x2. It can be shown that the dual number 6 ± h0 (except in the degenerate case when h0h0† = 0) has exactly one of ±the four hyperbolic Euler forms (Sobczyk, 1995),

ρ exp(Iφ) h0 = ± , ± ρI exp(Iφ) ± where ρ = h h † and φ is the hyperbolic angle deﬁned by h . Only in q| 0 0 | ± 0 the case when the sign of h0 can be chosen such that h0 = ρ exp(Iφ), will h have four well-deﬁned± square roots in ID. For± this case we have ± 0 k k 1 D = = exp( Iφ), where k = 1, I. (91) √ h √ρ −2 ± ± ± 0 Once we have found D, we also have A, B and C (88)

2 1 kh (h0) 2 hh00 h00 B = ,A = k − 3 ,C = k 3 , √ 2 − 2 h0 (h0) 2(h0) but it is not, in general, possible to solve for the transformation h(zx) 1 which corresponds to a given Ω0 = a− e g(zx) dzx. However, we can ﬁnd g(zx) in terms of the function h(zx): From (85) and (86), we obtain the second order diﬀerential equation for the conformal h-twistor of G,

B00 1 B = g(zx) . (92) D00 −2 D From (92) and (90) or (91), we have

2 D00 h000 3 h00 g(zx) = 2 = . − D h0 − 2 h0

It is recognized that g(zx) is the Schwarzian derivative of h(zx), which vanishes whenever h(zx) is a M¨obiustransformation. There are many possibilities for the further study of the Schwarzian derivative and its generalizations (Kobayashi and Wada, 2000).

Acknowledgements

Jos´ePozo acknowledges the support of the Spanish Ministry of Edu- cation (MEC), grant AP96-52209390, the project PB96-0384, and the Catalan Physics Society (IEC). Garret Sobczyk gratefully acknowledges the support of INIP of the Universidad de Las Americas-Puebla, and CIMAT-Guanajuato during his Sabbatical, Fall 1999.

gjfinPDF.tex; 6/02/2002; 14:45; p.41 42 J. Pozo and G. Sobczyk

References

R. Ablamowicz, and N. Salingaros. On the Relationship Between Twistors and Clifford Algebras. Letters in Mathematical Physics, 9, 149-155, 1985. Clifford Algebras and their Applications in Mathematical Physics. Editors: A. Ablamowicz, and B. Fauser. Birkäser,Boston (2000). M. Barnabei, A. Brini, and G.C. Rota. On the exterior calculus of invariant theory. J. Algebra, 96:120–160, 1985. Geometric Algebra with Applications in Science and Engineering. Editors: E. Bayro Corrochano, and G. Sobczyk. Birkäser,Boston. J. Cnops (1996) Vahlen Matrices for Non-definite Metrics. in Clifford Algebras with Numeric and Symbolic Computations. Editors: R. Ablamowicz, P. Lounesto, J.M. Parra. Birkhäuser,Boston. P. J. Davis. The Schwarz Function and its Applications. The Mathematical Association of America, 1974. C. Doran, D. Hestenes, F. Sommen, and N. Van Acker. Lie groups as spin groups. J. Math. Phys., 34:3642-3669, 1993. A. W. M. Dress and T. F. Havel. Distance Geometry and Geometric Algebra. Foundations of Physics, 23(10):1357–1374, 1993. W. Fulton and J. Harris. Representation Theory: A First Course. Springer-Verlag, 1991. T. Havel. Geometric Algebra and Mobius Sphere Geometry as a Basis for Eu- clidean Invariant Theory. Editor: N.L. White, Invariant Methods in Discrete and Computational Geometry: 245–256, Kluwer 1995. J. Haantjes. Conformal representations of an n-dimensional euclidean space with a non-definite fundamental form on itself. Proc. Ned. Akad. Wet (Math), 40, 700–705, 1937. D. Hestenes. The Design of Linear Algebra and Geometry. Acta Applicandae Mathematicae 23:65–93, Kluwer Academic Publishers, 1991. D. Hestenes and G. Sobczyk. Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics. D. Reidel, Dordrecht, 1984. D. Hestenes, and R. Ziegler (1991). Projective geometry with Clifford algebra. Acta Applicandae Mathematicae, 23: 25–63. O. Kobayashi and M. Wada. The Schwarzian and MöbiusTransformations in Higher Dimensions 5th International Clifford Algebra Conference, Ixtapa 1999. In: Clifford Algebras and their Applications in Mathematical Physics, Vol. 2, Editors J. Ryan and W. Sprösig,Birkhäuser2000. H. Li, D. Hestenes, and A. Rockwood. Generalized Homogeneous Coordinates for Computational Geometry. To be published P. Lounesto. CLICAL software paquet and user manual. Helsinki University of Technology of Mathematics, Research Report A248, 1994. P. Lounesto. Clifford Algebras and Spinors. Cambridge University Press, Cambridge, 1997. P. Lounesto and A. Springer. Mobius Transformations and Clifford Algebras of Euclidean and Anti-Euclidean Spaces. J. Lawrynowicz (ed.), Deformations of Mathematical Structures, 79-90, Kluwer Academic Publishers, 1989. J. G. Maks. Modulo (1,1) Periodicity of Clifford Algebras and Generalized Mobius Transformations. Technical University of Delft (Dissertation), 1989. Z. Nehari. Conformal Mapping McGraw-Hill, 1952. R. Penrose and M.A.H. MacCallum. Twistor Theory: an Approach to Quantization of Fields and Space-time. Phys. Rep., 6C, 241-316, 1972.

gjfinPDF.tex; 6/02/2002; 14:45; p.42 Geometric Algebra in Linear Algebra and Geometry 43

I. R. Porteous. Clifford Algebras and the Classical Groups Cambridge University Press, 1995. C. R. Rao, and S. K. Mitra. Generalized Inverse of Matrices and its Applications John Wiley & Sons, Inc., New York 1971. G. Sobczyk (1993). Jordan Form In Associative Algebras. In: Clifford Alge- bras and their Applications in Mathematical Physics, Proceedings of the Third International Clifford Workshop Edited by Fred Brackx, Kluwer, Dordrect, pp.33-41. G. Sobczyk. The Hyperbolic Number Plane. The College Mathematics Journal, 268-280:Vol. 26, No. 4 (1995). G. Sobczyk. Structure of Factor Algebras and Clifford Algebra. Linear Algebra and its Applications, 241-243:803-810 (1996). G. Sobczyk. The Generalized Spectral Decomposition of a Linear Operator. The College Mathematics Journal, 27–38, Vol. 28, No. 1 (1997). G. Sobczyk. Spectral integral domains in the classroom. Aportaciones Matematicas, Serie Comunicaciones 20 (1997) 169–188. G. Sobczyk. The Missing Spectral Basis in Algebra and Number Theory. The American Mathematical Monthly, 108, No. 4, (2001) 336–346. J. W. Young. Projective Geometry The Open Court Publishing Company, Chicago, ILL, 1930.

gjfinPDF.tex; 6/02/2002; 14:45; p.43 gjfinPDF.tex; 6/02/2002; 14:45; p.44