Geometric Algebra in Linear Algebra and Geometry 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 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, M¨obiustransformation, 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 in- terest" 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 geo- metric 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, repre- sent 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 ( ). ThisN 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 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 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 com- pactly formulated. Examples of calculations are given in the affine plane which are just plain fun! Chapter 4 explores the deep relationships which exist between pro- jective 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 introduc- tion 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) f g ··· 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 matrixf g 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 f g 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 off 1-formsg over the the field , and let e Nbe the dual basis of with respect to the basis e of .K If we nowf interpretg each of the vectorsN in e to be the row vectorsf g ofN the standard basis of the identity matrix idf(ng) of the n n matrix algebra ( ), we can again make the identification e =×id(n). Because we wishM K to be able to interpret the elements of ef gas row vectors, we will always write the vectors in e in the columnf g vector form f g e1 0 e2 1 e = B C (2) f g B · C B · C @ en A We also assume that the column vector e obeys all the rules of matrix addition and multiplication of a n 1 columnf g 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 2 N x1 0 x2 1 n x = e x e = ( e1 e2 en ) = xiei (3) f g f g ·· B · C X B C i=1 B · C @ xn A for xi IR, where 2 x1 0 x2 1 x e = B C f g B · C B · C @ xn A are the column vector of components of the vector x with respect to the basis e . Since vectorsf g in are represented by column vectors, and vectors y by row vectors,N we define the operation of transpose of the vector x 2byN e1 0 e2 1 t t t x = ( e x e ) = x e e = ( x1 x2 : : : xn ) B C (4) f g f g f gf g B · C B · C @ en A In the case of the complex field = , we have K C t x1 0 x2 1 x∗e = ( x1 x2 xn ) = B C (5) f g ··· B · C B · C @ xn A The transpose and Hermitian transpose operations allows us to move between the reciprocal vector spaces and .