Clifford Algebra Inverse and Determinant Inverse and Determinant in 0 to 5 Dimensional Clifford Algebra Peruzan Dadbeha) (Dated: March 15, 2011) This paper presents equations for the inverse of a Clifford number in Clifford algebras of up to five dimensions. In presenting these, there are also presented formulas for the determinant and adjugate of a general Clifford number of up to five dimensions, matching the determinant and adjugate of the matrix representations of the algebra. These equations are independent of the metric used. PACS numbers: 02.40.Gh, 02.10.-v, 02.10.De Keywords: clifford algebra, geometric algebra, inverse, determinant, adjugate, cofactor, adjoint I. INTRODUCTION the grade. The possible grades are from the grade-zero scalar and grade-1 vector elements, up to the grade-(d−1) Clifford algebra is one of the more useful math tools pseudovector and grade-d pseudoscalar, where d is the di- for modeling and analysis of geometric relations and ori- mension of Gd. entations. The algebra is structured as the sum of a Structurally, using the 1–dimensional ei’s, or 1-basis, scalar term plus terms of anticommuting products of vec- to represent Gd involves manipulating the products in tors. The algebra itself is independent of the basis used each term of a Clifford number by using the anticom- for computations, and only depends on the grades of muting part of the Clifford product to move the 1-basis the r-vector parts and their relative orientations. It is ei parts around in the product, as well as using the inner- this implementation (demonstrated via the standard or- product to eliminate the ei pairs with their metric equiv- thonormal representation) that will be used in the proofs. alent, so that one is left with terms of Clifford products of Inverses of Clifford numbers in 3–dimensions unique ei’s. Table (I) lists the additional r-basis elements and higher have been computed by using matrix for each subsequent dimension after the zero-dimensional 2 representations , since self-contained Clifford algebra G0 scalar element, here given the label e0 with the under- formulations for these inverses have been missing. This standing that e0 = 1. Counting the basis in the Table (I), paper presents general inverse expressions in Clifford a Clifford algebra of dimension d has 2d unique r-basis algebras of up to 5–dimensions. elements. A general Clifford algebra element is referred to as a Clifford number or a multivector, and can be written as a 1 A. Clifford Algebra Basics sum of the r-basis elements from Table (I) with real (com- plex) coefficients. From this representation, a Clifford The Clifford algebra of d-dimensions, Gd, is an ex- number can be separated into a sum of various grades. tended vector algebra over the real (or complex) num- The sum of the components of a specific grade-n is then bers. The algebraic representation can be generated by called an n-vector. The n-vector parts are written as the the orthonormal bases vectors {ei, i =1, ..., d} of the reg- sum of its n-basis components, Xi...ei..., which will be ular vector space Vd. For a vector space with a diagonal referred to as its r-components. It is these r-components metric {g11,g22,...,gdd}, the Clifford product of the ba- and their corresponding products that form the founda- sis 1-vectors satisfy the fundamental product relation: tion of the proofs and discussions to follow. Additionally, one can also distinguish an n-blade as an arXiv:1104.0067v2 [math-ph] 16 Aug 2012 e e = e · e + e ∧ e = g δ + e (1) i j i j i j ij i j ij n-vector for which there exists an alternative basis with ′ Where the inner-product, ei · ej , commutes and matches basis-element ei... in which the n-vector can be written as ′ the standard inner-product of two vectors. The discus- one term xei.... Alternatively, for a given n-vector Xn, if sion here is for a diagonal metric, although the equations there is a single basis element ei... for which ei...Xn results can be generalized to non-diagonal metrics. in a 1-vector, then the n-vector Xn is also an n-blade. The outer-product of two 1-vectors anticommute: This is because, via linear algebra, any combination of ei∧ej = −ej∧ei, and is non-zero for i6=j. The additional 1-vectors is also a 1-blade. Pseudovectors and pseudo- elements of the basis for Gd are generated by the sub- scalars can be referred to as blades, while scalars are not sequent outer-products of the ei basis vectors. As an since scalars have no basis. For all other non-zero grades, example, e12 = e1e2 = e1∧e2 is one of the grade-2 basis an n-vector is generally not an n-blade. elements, or 2-basis. The number of ei in a general ba- For example, in 4–dimensions, the 2-vectors can sis element after reduction in the product is defined as be separated into two different 2-blades: the space 2-blade (xe23 + ye13 + ze12) and the time 2-blade (ue14 + ve24 + we34) used for rotations and boosts in Spe- cial Relativity. Here, the space 2-blade times e123 results a)email: [email protected] in a 1-vector, while the time part requires multiplication Clifford Algebra Inverse and Determinant 2 ple, the 3–dimensional Clifford number, TABLE I. Standard r-Basis Elements dim grade-1 grade-2 grade-3 grade-4 grade-5 #basis X = a0 + a1e1 + a2e2 + a12e12 + (6) 0 e (†) 1 0 a3e3 + a13e13 + a23e23 + a123e123 1 e1 2 2 e2 e12 4 can be written in the complex form, with I = e123, as, 3 e3 e13,e23 e123 8 X = (a0 + Ia123)e0 + (a1 + Ia23)e1 + (7) 4 e4 e14,e24 e124,e134 e1234 16 e34 e234 (a2 − Ia13)e2 + (a12 − Ia3)e12 5 e5 e15,e25 e125,e135 e1235,e1245 e12345 32 In even dimensions, it is important to maintain the or- e35,e45 e145,e235 e1345,e2345 der of the pseudoscalar token I and the basis elements e245,e345 ei..., since the pseudoscalar will anticommute with odd (†)The scalar element, e =1, is grade-zero. 0 grades. The net sign change for commuting a pseudo- scalar with an r-vector is (−1)r(d−1). In odd dimen- by e4 to show it is a 2-blade. In 2 and 3–dimensions, the sions, maintaining product order is not needed since the 2-vectors are pseudoscalars and pseudovectors, respec- pseudoscalar commutes with all grades. The noncommu- tively, and are therefore 2-blades. Blades will be impor- tivity of the even-dimensional pseudoscalar is the main tant in the special-case determinants discussed later. obstacles in constructing even-dimensional complex de- The pseudoscalar e1...d is often given importance in its terminants from those of the previous odd dimension. analogy to the imaginary ı of the Complex numbers. For Another important difference between even and odd this reason, it is often given the label I, although its self- dimensions is in the middle grades. For even dimen- product is not necessarily (−1), but instead given by the sions, there is a middle grade “d/2” which is split between self-product the real and imaginary parts, such as the 1-vector part (a1 − Ia2) of Equation (5). In 4–dimensions, it is the 2 d(d−1)/2 I = e1...de1...d = (−1) g11g22. .gdd (2) space and time bivector parts that become the real and imaginary parts of the complex bivector. This would sug- where it takes d(d−1)/2 anticommutations to reverse the gest that the even-dimensional complex representation is product order in the pseudoscalar from e1...d to ed...1. inherently representation dependent, however, splitting The final value of the pseudoscalar squared depends on the 2-vectors evenly into non-blade half real and half the metric values gii. Clifford algebras usually have a imaginary parts shows that the complex representation diagonal metric with values of +1, −1 and zero. Such a (r,s,t) can be structured as representation independent. Clifford algebra is labeled Gd , where {r,s,t} are the numbers of +1, −1 and zero metric elements respectively. The most common metrics are the Euclidean {1,..., 1} B. The 5–Dimensional Self-Product (d,0,0) for Gd , and the Minkowski metric {1, −1,..., −1} for (1,d−1,0) A general Clifford number in 5–dimensions has 32 com- Gd . The importance of the unit pseudoscalar I in this pa- ponents: one scalar 0-component, five 1-components, ten per is in writing a Clifford number in its complex rep- 2-components, ten 3-components, five 4-components and resentation. In the complex representation, the unit one 5-component, as listed in Table (I). When a general pseudoscalar takes on the role of the imaginary token, al- multivector is multiplied by itself, the contributions to though it does not necessarily square to −1. The complex the grades in the result are based on the sign changes representation is obtained by first multiplying the Clif- resulting from the left and right products of the various ford number by the unit pseudoscalar, a product called r-components. Those products that result in the same the dual or left-dual, dual[X]= IX. To include dimen- sign will commute and contribute, while those that have sions in which I2 = −1, the procedure requires multiply- opposite signs will anticommute and cancel. ing by I4 = 1, with I3 taking X to its third dual.