DOCSLIB.ORG

Functions of Multivectors in 3D Euclidean Geometric Algebra Via Spectral Decomposition (For Physicists and Engineers)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Functions of Multivectors in 3D Euclidean Geometric Algebra Via Spectral Decomposition (For Physicists and Engineers) Functions of multivectors in 3D Euclidean geometric algebra via spectral decomposition (for physicists and engineers) Miroslav Josipović [email protected] December, 2015 Geometric algebra is a powerful mathematical tool for description of physical phenomena. The article [ 3] gives a thorough analysis of functions of multivectors in Cl 3 relaying on involutions, especially Clifford conjugation and complex structure of algebra. Here is discussed another elegant way to do that, relaying on complex structure and idempotents of algebra. Implementation of Cl 3 using ordinary complex algebra is briefly discussed. Keywords: function of multivector, idempotent, nilpotent, spectral decomposition, unipodal numbers, geometric algebra, Clifford conjugation, multivector amplitude, bilinear transformations 1. Numbers Geometric algebra is a promising platform for mathematical analysis of physical phenomena. The simplicity and naturalness of the initial assumptions and the possibility of formulation of (all?) main physical theories with the same mathematical language imposes the need for a serious study of this beautiful mathematical structure. Many authors have made significant contributions and there is some surprising conclusions. Important one is certainly the possibility of natural defining Minkowski metrics within Euclidean 3D space without introduction of negative signature, that is, without defining time as the fourth dimension ([1, 6]). In Euclidean 3D space we define orthogonal unit vectors e1, e 2, e 3 with the property e2 = 1 ee+ ee = 0 i , ij ji , so one could recognize the rule for multiplication of Pauli matrices. Non-commutative product of two vectors is ab= a ⋅ b + a ∧ b , sum of symmetric (inner product) and anti- symmetric part (wedge product). Each element of the algebra (Cl 3) can be expressed as linear combination of elements of 23 – dimensional basis (Clifford basis ) e e e ee ee ee eee {1, 1, 2, 3, 12, 31, 23, 123 }, where we have a scalar, three vectors, three bivectors and pseudoscalar. According to the number of unit vectors in the product we are talking about odd or even elements. If we define = e e e j 1 2 3 it is easy to show that pseudoscalar j has two interesting properties in Cl 3: 1) j2 = − 1, 2) jX= Xj , for any element X of algebra, and behaves like an ordinary imaginary unit , which enables as to study a rich complex structure of Cl 3. This property we have for n = 3, 7, … [3]. Bivectors can be expressed as product of unit pseudoscalar and vectors, jv . 1 We define a general element of the algebra (multivector ) Mt=+++=+xn jjbz F, ztjb =+, Fxn =+ j where z is a complex scalar and an element of center of algebra , while F , by analogy, is a ∗ ∗ complex vector . Complex conjugation ( j ∈ ℂ ) is z= z† = t − jb , F= F† = x − j n , where dagger means reversion , to be defined later in the text. The complex structure allows a different ways of expressing multivectors, one is Mt=++xn j + jbtj =+ n + jbj( − x ) , where multivector of the form a+ vj vˆ belongs to the even part of the algebra and can be associated with rotations, spinors or quaternions. Also we could treat multivector as ([ 12]) 3 M =α + αe α ∈ ℂ 0 ∑ i i, k and implement it relying on ordinary complex numbers. i=1 Main involutions in Clifford algebra are: ˆ 1) grade involution: Mt=−x + j n − jb ∗ ∗ 2) reverse (adjoint ): M† =+− tx j n − jb = z + F 3) Clifford conjugation: Mt=−−xn j + jbz =+ F =− z F , where an asterisk stands for a complex conjugate. Grade involution is the transformation ˆ x= − x (space inversion ), while reverse in Cl 3 is like a complex conjugation, x†= x , j † = − j . Clifford conjugation is combination of two involutions MM=ˆ † , x =− x , jj = . Bivectors given as a wedge product could be expressed as x∧ y =j xy × , where x× y is a cross product . An application of involutions is easy now. 2 Defining paravector p= t + x we have ppt=+x =+()() t x t − x =− tx2 2 and we have a usual metric of special relativity. From MM=ˆ ⇒ Mtj =+ n , the even part of the algebra (spinors ). † From MM=† ⇒ Mt =+ x , paravector; reverse is an anti-automorphism (MM†) = MM † , so MM † (square of multivector magnitude , [ 2]) is a paravector. From M= M ⇒ M =+ t jbz = , a complex scalar. Clifford conjugation is anti-automorphism, MM= MM , so MM (square of multivector amplitude , [ 2]) is a complex scalar and there is no other “amplitude” with such a property ([ 1]). We define a multivector amplitude M (hereinafter MA) 2 2 2 2 2 MMM= =−+−+ txnb2 jtb() −⋅x n , M ∈ ℂ (3) which we could express as MMMz==+()()FF z −=− z2 FF 2, 2 =−+⋅∈ xnj 22 2 xn ℂ . 2 2 2 2 2 For F=0, or F = N = 0 fallows M= z ( N=x − n , x ⋅= n 0 is a nilpotent in the F 2 =c ∈ ℝ x⋅ n = 0 F F 2 ∈ ℂ algebra). For ( , whirl [1]) here is used a designation ()c . From 2 F= F F2 F 2 = FFˆ= −= F2 FF =− FF ˆ 2 =− we have ()1/ , () 1 1 , and / /j ()1 , 1 (complex = u=± uuuu2 = = unit vector ). With f F ()1 we also define ±(1f ) / 2, ±±+−, 0 , idempotents 2 of the algebra ([ 1, 9]). Every idempotent in Cl 3 can be expressed as u±= p ± +N ±±, N = 0 , =1 ± e / 2. where p± are simple idempotents, like p± ( 1 ) For example, u=1 +e + e + j e / 2, N= e + j e (figure below). + ( 1 2 3 ) 2 3 2. Implementation 3 M=+α ααe =+ A α ∈ ℂ From 0∑ i i 0 , k , it is i=1 easy to implement the algebra on computer using ordinary complex numbers only. In [15 ] are defined products: 3 A B = ∑αi β i (generalized inner product ), i=1 A⊗ B = e det[ i , α i, β i ], (generalized outer product ), and AB= AB + A ⊗ B (generalized geometric product ). α+A α += B ααα +++ ABAB α α Now we have ( A)( B) ABB A . We can find i for multivector Mt=+x + j n + jb using linear independency. 3. Spectral decomposition F 2 = 1 Starting from multivector (1) 2 2 "whirl" Mz=+=+F z Ff =+ xy fF, ≠ 0 we see the form of unipodal-like numbers. Defining M± = x ± y and recalling a relation (easy to proof) fu±= ± u ± follows =± =± = Mu±( xyuf ) ±( xyu) ±±± Mu , so we have a projection. Spectral basis u± is very useful because the binomial expansion of a multivector is very simple M2 = Mu + Mu2 =+⇒=+ Mu2 Mu 2 Mn Mu n Mun n ∈ ℤ ()++ −− ++ −− ++ −− , , where n < 0 is possible for M ≠ 0 ( M−1 = M/ ( MM ) ). 3 − Defining conjugation ()ab+f = ab − f (obviously the Clifford conjugation) we have − ()()ab+f ab +=− f ab2 2 , where a2− b 2 = MM− = MM is a multivector amplitude. In u− = u MM− =+ Mu Mu Mu += Mu MM spectral basis using ± ∓ we have ( ++ −−)( +− −+) +− . Starting from M= z + F we have 2 x y 2 2 M=+ zFff =+=+= x y ρ f ρϕ()cosh + f sinh ϕρ , =−x y , ρ ρ obtaining the polar form of a multivector. If a multivector amplitude is zero we have light-like multivector and there is no a polar form. Now defining tanh ϕ= ϑ (“velocity”) we have M =ρ()()cosh ϕ +f sinh ϕργϑγ =+ 1 f , −1 =−1 ϑ 2 and in the spectral basis M=1 + = kukuk + ⇒= 1 ±= K ±1 ργϑ( f ) ++ −− ± ργϑρ( ) , where K =()()1 +ϑ / 1 − ϑ is a generalized Bondi factor (ϕ = log K ). Now we have Ku+ Ku−1 Ku += Ku − 1 KKu + KKu −− 11 =+⇒= Ku Ku − 1 K KK ( 1+−+− 1)( 1 1) 1 2 + 1 2 −+− 1 2 , or 1+ 1 += 1 ++ += γγ12( ϑ 1f)( ϑ 2 f) γγ 12( ( ϑϑ 12) f ϑϑ 12 ) 1+ 1 ++ /1 + ⇒ γγ12()()() ϑϑ 12() ϑϑ 12 ϑϑ 12 f = + =+ + γγγ12(1 ϑϑ 12) , ϑ( ϑϑ 12) /1( ϑϑ 12 ) , and we have “velocity addition rule”. So, every multivector could be mathematically treated like an ordinary boost in special relativity. For ρ = 1 we have a “boost” 1+ =Ku + K−1 u γ( ϑ f ) + − as transformation that preserves multivector amplitude and should be considered as a part of the Lorentz group ([1, 2]). As a simple example of a unit complex vector we already mentioned fee=++1 2j e 3 =++ ee 1 2 ee 12 , completely in Cl 2, suggesting F that one could analyze problem in basis (1, (1) ) or related spectral basis in e1 e 2 plane and rotate all elements to obtain relations for an arbitrary orientation of a plane, using powerful apparatus of geometric algebra for rotations. Mapping basis (1, f ) to (eφf , f e φ f ) we obtain new orthogonal basis and new components of multivector ab+→f ae′φf + be ′ f φ f =+( abe ′′ f ) φ f , aae′=−φf, bbe ′ = − φ f , ab +=+f ab ′′ f . 4. Functions of multivectors Using series expansion it is straight forward to find a closed formulae for (analytic at least) functions. If F 2 = 0 we have fM( ) = fz( ) and it is easy to find closed form using theory of functions on the complex field. Otherwise, from the series expansion 4 n f( ) (0) x n f()() x= f 0 + ∑ n n! 2 n n using M= Mu++ + Mu −− we have = + fM( ) fMu( ++) fMu( −−) 2 and, again, it is “easy” to find a closed form because of M ± ∈ ℂ . For M =F = F f we have =±⇒F2 = F 2 +− F 2 . If function is even we have M± fMf() ( ) uf+( ) u − F= F2 + = F 2 and similarly for odd functions f() f( )() uuf+ − ( ) =2 − = 2 . For M=+ z F, F2 = N 2 = 0 there is no spectral f()FF f( )() uuf+ − ( Ff) nn n n −1 decomposition ( f is not defined), but we have M=+() zN = znz + N , giving fz( +N) = fz( ) + fz′( ) N .
Load more

Recommended publications
  • Quaternions and Cli Ord Geometric Algebras
    Quaternions and Cliord Geometric Algebras Robert Benjamin Easter First Draft Edition (v1) (c) copyright 2015, Robert Benjamin Easter, all rights reserved. Preface As a rst rough draft that has been put together very quickly, this book is likely to contain errata and disorganization. The references list and inline citations are very incompete, so the reader should search around for more references. I do not claim to be the inventor of any of the mathematics found here. However, some parts of this book may be considered new in some sense and were in small parts my own original research. Much of the contents was originally written by me as contributions to a web encyclopedia project just for fun, but for various reasons was inappropriate in an encyclopedic volume. I did not originally intend to write this book. This is not a dissertation, nor did its development receive any funding or proper peer review. I oer this free book to the public, such as it is, in the hope it could be helpful to an interested reader. June 19, 2015 - Robert B. Easter. (v1) [email protected] 3 Table of contents Preface . 3 List of gures . 9 1 Quaternion Algebra . 11 1.1 The Quaternion Formula . 11 1.2 The Scalar and Vector Parts . 15 1.3 The Quaternion Product . 16 1.4 The Dot Product . 16 1.5 The Cross Product . 17 1.6 Conjugates . 18 1.7 Tensor or Magnitude . 20 1.8 Versors . 20 1.9 Biradials . 22 1.10 Quaternion Identities . 23 1.11 The Biradial b/a .
    [Show full text]
  • 1 Parity 2 Time Reversal
    Even Odd Symmetry Lecture 9 1 Parity The normal modes of a string have either even or odd symmetry. This also occurs for stationary states in Quantum Mechanics. The transformation is called partiy. We previously found for the harmonic oscillator that there were 2 distinct types of wave function solutions characterized by the selection of the starting integer in their series representation. This selection produced a series in odd or even powers of the coordiante so that the wave function was either odd or even upon reflections about the origin, x = 0. Since the potential energy function depends on the square of the position, x2, the energy eignevalue was always positive and independent of whether the eigenfunctions were odd or even under reflection. In 1-D parity is the symmetry operation, x → −x. In 3-D the strong interaction is invarient under the symmetry of parity. ~r → −~r Parity is a mirror reflection plus a rotation of 180◦, and transforms a right-handed coordinate system into a left-handed one. Our Macroscopic world is clearly “handed”, but “handedness” in fundamental interactions is more involved. Vectors (tensors of rank 1), as illustrated in the definition above, change sign under Parity. Scalars (tensors of rank 0) do not. One can then construct, using tensor algebra, new tensors which reduce the tensor rank and/or change the symmetry of the tensor. Thus a dual of a symmetric tensor of rank 2 is a pseudovector (cross product of two vectors), and a scalar product of a pseudovector and a vector creates a pseudoscalar. We will construct bilinear forms below which have these rotational and reflection char- acteristics.
    [Show full text]
  • A New Description of Space and Time Using Clifford Multivectors
    A new description of space and time using Clifford multivectors James M. Chappell† , Nicolangelo Iannella† , Azhar Iqbal† , Mark Chappell‡ , Derek Abbott† †School of Electrical and Electronic Engineering, University of Adelaide, South Australia 5005, Australia ‡Griffith Institute, Griffith University, Queensland 4122, Australia Abstract Following the development of the special theory of relativity in 1905, Minkowski pro- posed a unified space and time structure consisting of three space dimensions and one time dimension, with relativistic effects then being natural consequences of this space- time geometry. In this paper, we illustrate how Clifford’s geometric algebra that utilizes multivectors to represent spacetime, provides an elegant mathematical framework for the study of relativistic phenomena. We show, with several examples, how the application of geometric algebra leads to the correct relativistic description of the physical phenomena being considered. This approach not only provides a compact mathematical representa- tion to tackle such phenomena, but also suggests some novel insights into the nature of time. Keywords: Geometric algebra, Clifford space, Spacetime, Multivectors, Algebraic framework 1. Introduction The physical world, based on early investigations, was deemed to possess three inde- pendent freedoms of translation, referred to as the three dimensions of space. This naive conclusion is also supported by more sophisticated analysis such as the existence of only five regular polyhedra and the inverse square force laws. If we lived in a world with four spatial dimensions, for example, we would be able to construct six regular solids, and in arXiv:1205.5195v2 [math-ph] 11 Oct 2012 five dimensions and above we would find only three [1].
    [Show full text]
  • 21. Orthonormal Bases
    21. Orthonormal Bases The canonical/standard basis 011 001 001 B C B C B C B0C B1C B0C e1 = B.C ; e2 = B.C ; : : : ; en = B.C B.C B.C B.C @.A @.A @.A 0 0 1 has many useful properties. • Each of the standard basis vectors has unit length: q p T jjeijj = ei ei = ei ei = 1: • The standard basis vectors are orthogonal (in other words, at right angles or perpendicular). T ei ej = ei ej = 0 when i 6= j This is summarized by ( 1 i = j eT e = δ = ; i j ij 0 i 6= j where δij is the Kronecker delta. Notice that the Kronecker delta gives the entries of the identity matrix. Given column vectors v and w, we have seen that the dot product v w is the same as the matrix multiplication vT w. This is the inner product on n T R . We can also form the outer product vw , which gives a square matrix. 1 The outer product on the standard basis vectors is interesting. Set T Π1 = e1e1 011 B C B0C = B.C 1 0 ::: 0 B.C @.A 0 01 0 ::: 01 B C B0 0 ::: 0C = B. .C B. .C @. .A 0 0 ::: 0 . T Πn = enen 001 B C B0C = B.C 0 0 ::: 1 B.C @.A 1 00 0 ::: 01 B C B0 0 ::: 0C = B. .C B. .C @. .A 0 0 ::: 1 In short, Πi is the diagonal square matrix with a 1 in the ith diagonal position and zeros everywhere else.
    [Show full text]
  • Multivector Differentiation and Linear Algebra 0.5Cm 17Th Santaló
    Multivector differentiation and Linear Algebra 17th Santalo´ Summer School 2016, Santander Joan Lasenby Signal Processing Group, Engineering Department, Cambridge, UK and Trinity College Cambridge [email protected], www-sigproc.eng.cam.ac.uk/ s jl 23 August 2016 1 / 78 Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... Summary Overview The Multivector Derivative. 2 / 78 Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... Summary Overview The Multivector Derivative. Examples of differentiation wrt multivectors. 3 / 78 Functional Differentiation: very briefly... Summary Overview The Multivector Derivative. Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. 4 / 78 Summary Overview The Multivector Derivative. Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... 5 / 78 Overview The Multivector Derivative. Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... Summary 6 / 78 We now want to generalise this idea to enable us to find the derivative of F(X), in the A ‘direction’ – where X is a general mixed grade multivector (so F(X) is a general multivector valued function of X). Let us use ∗ to denote taking the scalar part, ie P ∗ Q ≡ hPQi. Then, provided A has same grades as X, it makes sense to define: F(X + tA) − F(X) A ∗ ¶XF(X) = lim t!0 t The Multivector Derivative Recall our definition of the directional derivative in the a direction F(x + ea) − F(x) a·r F(x) = lim e!0 e 7 / 78 Let us use ∗ to denote taking the scalar part, ie P ∗ Q ≡ hPQi.
    [Show full text]
  • A Clifford Dyadic Superfield from Bilateral Interactions of Geometric Multispin Dirac Theory
    A CLIFFORD DYADIC SUPERFIELD FROM BILATERAL INTERACTIONS OF GEOMETRIC MULTISPIN DIRAC THEORY WILLIAM M. PEZZAGLIA JR. Department of Physia, Santa Clam University Santa Clam, CA 95053, U.S.A., [email protected] and ALFRED W. DIFFER Department of Phyaia, American River College Sacramento, CA 958i1, U.S.A. (Received: November 5, 1993) Abstract. Multivector quantum mechanics utilizes wavefunctions which a.re Clifford ag­ gregates (e.g. sum of scalar, vector, bivector). This is equivalent to multispinors con­ structed of Dirac matrices, with the representation independent form of the generators geometrically interpreted as the basis vectors of spacetime. Multiple generations of par­ ticles appear as left ideals of the algebra, coupled only by now-allowed right-side applied (dextral) operations. A generalized bilateral (two-sided operation) coupling is propoeed which includes the above mentioned dextrad field, and the spin-gauge interaction as partic­ ular cases. This leads to a new principle of poly-dimensional covariance, in which physical laws are invariant under the reshuffling of coordinate geometry. Such a multigeometric su­ perfield equation is proposed, whi~h is sourced by a bilateral current. In order to express the superfield in representation and coordinate free form, we introduce Eddington E-F double-frame numbers. Symmetric tensors can now be represented as 4D "dyads", which actually are elements of a global SD Clifford algebra.. As a restricted example, the dyadic field created by the Greider-Ross multivector current (of a Dirac electron) describes both electromagnetic and Morris-Greider gravitational interactions. Key words: spin-gauge, multivector, clifford, dyadic 1. Introduction Multi vector physics is a grand scheme in which we attempt to describe all ba­ sic physical structure and phenomena by a single geometrically interpretable Algebra.
    [Show full text]
  • Lecture 4: April 8, 2021 1 Orthogonality and Orthonormality
    Mathematical Toolkit Spring 2021 Lecture 4: April 8, 2021 Lecturer: Avrim Blum (notes based on notes from Madhur Tulsiani) 1 Orthogonality and orthonormality Definition 1.1 Two vectors u, v in an inner product space are said to be orthogonal if hu, vi = 0. A set of vectors S ⊆ V is said to consist of mutually orthogonal vectors if hu, vi = 0 for all u 6= v, u, v 2 S. A set of S ⊆ V is said to be orthonormal if hu, vi = 0 for all u 6= v, u, v 2 S and kuk = 1 for all u 2 S. Proposition 1.2 A set S ⊆ V n f0V g consisting of mutually orthogonal vectors is linearly inde- pendent. Proposition 1.3 (Gram-Schmidt orthogonalization) Given a finite set fv1,..., vng of linearly independent vectors, there exists a set of orthonormal vectors fw1,..., wng such that Span (fw1,..., wng) = Span (fv1,..., vng) . Proof: By induction. The case with one vector is trivial. Given the statement for k vectors and orthonormal fw1,..., wkg such that Span (fw1,..., wkg) = Span (fv1,..., vkg) , define k u + u = v − hw , v i · w and w = k 1 . k+1 k+1 ∑ i k+1 i k+1 k k i=1 uk+1 We can now check that the set fw1,..., wk+1g satisfies the required conditions. Unit length is clear, so let’s check orthogonality: k uk+1, wj = vk+1, wj − ∑ hwi, vk+1i · wi, wj = vk+1, wj − wj, vk+1 = 0. i=1 Corollary 1.4 Every finite dimensional inner product space has an orthonormal basis.
    [Show full text]
  • Geometric-Algebra Adaptive Filters Wilder B
    1 Geometric-Algebra Adaptive Filters Wilder B. Lopes∗, Member, IEEE, Cassio G. Lopesy, Senior Member, IEEE Abstract—This paper presents a new class of adaptive filters, namely Geometric-Algebra Adaptive Filters (GAAFs). They are Faces generated by formulating the underlying minimization problem (a deterministic cost function) from the perspective of Geometric Algebra (GA), a comprehensive mathematical language well- Edges suited for the description of geometric transformations. Also, (directed lines) differently from standard adaptive-filtering theory, Geometric Calculus (the extension of GA to differential calculus) allows Fig. 1. A polyhedron (3-dimensional polytope) can be completely described for applying the same derivation techniques regardless of the by the geometric multiplication of its edges (oriented lines, vectors), which type (subalgebra) of the data, i.e., real, complex numbers, generate the faces and hypersurfaces (in the case of a general n-dimensional quaternions, etc. Relying on those characteristics (among others), polytope). a deterministic quadratic cost function is posed, from which the GAAFs are devised, providing a generalization of regular adaptive filters to subalgebras of GA. From the obtained update rule, it is shown how to recover the following least-mean squares perform calculus with hypercomplex quantities, i.e., elements (LMS) adaptive filter variants: real-entries LMS, complex LMS, that generalize complex numbers for higher dimensions [2]– and quaternions LMS. Mean-square analysis and simulations in [10]. a system identification scenario are provided, showing very good agreement for different levels of measurement noise. GA-based AFs were first introduced in [11], [12], where they were successfully employed to estimate the geometric Index Terms—Adaptive filtering, geometric algebra, quater- transformation (rotation and translation) that aligns a pair of nions.
    [Show full text]
  • Determinants in Geometric Algebra
    Determinants in Geometric Algebra Eckhard Hitzer 16 June 2003, recovered+expanded May 2020 1 Definition Let f be a linear map1, of a real linear vector space Rn into itself, an endomor- phism n 0 n f : a 2 R ! a 2 R : (1) This map is extended by outermorphism (symbol f) to act linearly on multi- vectors f(a1 ^ a2 ::: ^ ak) = f(a1) ^ f(a2) ::: ^ f(ak); k ≤ n: (2) By definition f is grade-preserving and linear, mapping multivectors to mul- tivectors. Examples are the reflections, rotations and translations described earlier. The outermorphism of a product of two linear maps fg is the product of the outermorphisms f g f[g(a1)] ^ f[g(a2)] ::: ^ f[g(ak)] = f[g(a1) ^ g(a2) ::: ^ g(ak)] = f[g(a1 ^ a2 ::: ^ ak)]; (3) with k ≤ n. The square brackets can safely be omitted. The n{grade pseudoscalars of a geometric algebra are unique up to a scalar factor. This can be used to define the determinant2 of a linear map as det(f) = f(I)I−1 = f(I) ∗ I−1; and therefore f(I) = det(f)I: (4) For an orthonormal basis fe1; e2;:::; eng the unit pseudoscalar is I = e1e2 ::: en −1 q q n(n−1)=2 with inverse I = (−1) enen−1 ::: e1 = (−1) (−1) I, where q gives the number of basis vectors, that square to −1 (the linear space is then Rp;q). According to Grassmann n-grade vectors represent oriented volume elements of dimension n. The determinant therefore shows how these volumes change under linear maps.
    [Show full text]
  • 28. Exterior Powers
    28. Exterior powers 28.1 Desiderata 28.2 Definitions, uniqueness, existence 28.3 Some elementary facts 28.4 Exterior powers Vif of maps 28.5 Exterior powers of free modules 28.6 Determinants revisited 28.7 Minors of matrices 28.8 Uniqueness in the structure theorem 28.9 Cartan's lemma 28.10 Cayley-Hamilton Theorem 28.11 Worked examples While many of the arguments here have analogues for tensor products, it is worthwhile to repeat these arguments with the relevant variations, both for practice, and to be sensitive to the differences. 1. Desiderata Again, we review missing items in our development of linear algebra. We are missing a development of determinants of matrices whose entries may be in commutative rings, rather than fields. We would like an intrinsic definition of determinants of endomorphisms, rather than one that depends upon a choice of coordinates, even if we eventually prove that the determinant is independent of the coordinates. We anticipate that Artin's axiomatization of determinants of matrices should be mirrored in much of what we do here. We want a direct and natural proof of the Cayley-Hamilton theorem. Linear algebra over fields is insufficient, since the introduction of the indeterminate x in the definition of the characteristic polynomial takes us outside the class of vector spaces over fields. We want to give a conceptual proof for the uniqueness part of the structure theorem for finitely-generated modules over principal ideal domains. Multi-linear algebra over fields is surely insufficient for this. 417 418 Exterior powers 2. Definitions, uniqueness, existence Let R be a commutative ring with 1.
    [Show full text]
  • New Foundations for Geometric Algebra1
    Text published in the electronic journal Clifford Analysis, Clifford Algebras and their Applications vol. 2, No. 3 (2013) pp. 193-211 New foundations for geometric algebra1 Ramon González Calvet Institut Pere Calders, Campus Universitat Autònoma de Barcelona, 08193 Cerdanyola del Vallès, Spain E-mail : [email protected] Abstract. New foundations for geometric algebra are proposed based upon the existing isomorphisms between geometric and matrix algebras. Each geometric algebra always has a faithful real matrix representation with a periodicity of 8. On the other hand, each matrix algebra is always embedded in a geometric algebra of a convenient dimension. The geometric product is also isomorphic to the matrix product, and many vector transformations such as rotations, axial symmetries and Lorentz transformations can be written in a form isomorphic to a similarity transformation of matrices. We collect the idea Dirac applied to develop the relativistic electron equation when he took a basis of matrices for the geometric algebra instead of a basis of geometric vectors. Of course, this way of understanding the geometric algebra requires new definitions: the geometric vector space is defined as the algebraic subspace that generates the rest of the matrix algebra by addition and multiplication; isometries are simply defined as the similarity transformations of matrices as shown above, and finally the norm of any element of the geometric algebra is defined as the nth root of the determinant of its representative matrix of order n. The main idea of this proposal is an arithmetic point of view consisting of reversing the roles of matrix and geometric algebras in the sense that geometric algebra is a way of accessing, working and understanding the most fundamental conception of matrix algebra as the algebra of transformations of multiple quantities.
    [Show full text]
  • The Construction of Spinors in Geometric Algebra
    The Construction of Spinors in Geometric Algebra Matthew R. Francis∗ and Arthur Kosowsky† Dept. of Physics and Astronomy, Rutgers University 136 Frelinghuysen Road, Piscataway, NJ 08854 (Dated: February 4, 2008) The relationship between spinors and Clifford (or geometric) algebra has long been studied, but little consistency may be found between the various approaches. However, when spinors are defined to be elements of the even subalgebra of some real geometric algebra, the gap between algebraic, geometric, and physical methods is closed. Spinors are developed in any number of dimensions from a discussion of spin groups, followed by the specific cases of U(1), SU(2), and SL(2, C) spinors. The physical observables in Schr¨odinger-Pauli theory and Dirac theory are found, and the relationship between Dirac, Lorentz, Weyl, and Majorana spinors is made explicit. The use of a real geometric algebra, as opposed to one defined over the complex numbers, provides a simpler construction and advantages of conceptual and theoretical clarity not available in other approaches. I. INTRODUCTION Spinors are used in a wide range of fields, from the quantum physics of fermions and general relativity, to fairly abstract areas of algebra and geometry. Independent of the particular application, the defining characteristic of spinors is their behavior under rotations: for a given angle θ that a vector or tensorial object rotates, a spinor rotates by θ/2, and hence takes two full rotations to return to its original configuration. The spin groups, which are universal coverings of the rotation groups, govern this behavior, and are frequently defined in the language of geometric (Clifford) algebras [1, 2].
    [Show full text]