p 2 p 3 p y p z It will not be considered in the present context; the interested reader can find more details in [05].

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1 1. Geometrical vectors A geometrical vector p represents a point P in space. The point P is an abstraction that often, but not always, requires a representation. Vector representations are given wrt a reference frame (see below) If P R 3 (a 3D space) then p a R 3 = [ px p y p z ] a [ p1 p i is the vector i-th coordinate wrt the chosen reference frame R a. p 2 p 3 ] a Affine geometry To treat points as vectors in linear (vector) spaces implies the definition of a zero point (the origin), i.e., a point with particular privileged characteristics. Since, in many applications, this is not always the case, a particular geometry that is origin-free has been set up. It is called affine geometry and is defined on affine spaces. Affine geometry is at the base of projective geometry and perspective transforms, as well as homogeneous vectors. It will not be considered in the present context; the interested reader can find more details in [05]. 2. Physical vectors A physical vector QP represents a physical quantity, such as linear or angular velocity, gravitational acceleration, force, torque, etc. A physical vectors is an oriented segment (aka directed line segment), with an application point Q that can be free or constrained, a direction and a magnitude. QP may be reference-free (as, e.g., a physical entity independent from the way we represent it, and from an origin), but we are usually representing it by a vector v qp R 3 = p q = that is the difference ot two geometrical vectors [ ] p1 q 1 p 2 q 2 p 3 q 3 [ v1 v 2 v 3 ] p P q Q Example: the local gravitational acceleration has direction and magnitude that are absolute, since they do not depend on the reference frame chosen Physical vectors Physical vectors We represents physical vectors with an icon; the most used icon is an arrow. This icon is sometimes misleading, as we will see, considering the symmetry properties of the physical vectors. Polar vectors Polar vectors (see [01]) are physical vectors that are symmetrical wrt a reflection through a parallel plane, and are antisymmetrical wrt a reflection through a perpendicular plane. Examples of physical polar vectors are displacements, linear velocities and forces.

2 Figure 1: Physical vectors. Figure 2: The vector icon. Axial vectors Axial vectors are physical vectors that are antisymmetrical wrt a reflection through a parallel plane, and are symmetrical wrt a reflection through a perpendicular plane. Examples of physical axial vectors are angular velocities and torques. 3. Mathematical vectors Mathematical vectors are abstract entities belonging to vector spaces. Vector Space Given a field F = {F ;+, }, a vector space V(F), is the set of elements, called vectors, that satisfies the following axiomatic properties: Vector sum: the operation +, called vector sum, is defined so that {V(F);+} is a commutative (abelian) group; the identity element is called 0; v+0 = v Product by a scalar: For each α F and each v V(F), it exists a vector αv V(F); Vector space For each α,β F and each v,w V(F) the following relations hold true: associative property wrt product by a scalar: α(β v) = (αβ)v 2

3 Figure 3: The polar vector. Figure 4: The axial vector. existence of the identity wrt product by a scalar: 1(v) = v; v distributive property wrt vector sum: α(v+w) = αv+ αw distributive property wrt product by a scalar: (α + β)v = αv+ β v Linear functions and dual spaces Given two vector spaces, V(F) and U(F), both defined on the same field F, a function f : V U is linear, if, for every v,w V and λ F the following axioms are true: f(v+w) = f(v)+ f(w) = f v+ f w f(λ v) = λ f(v) = λ f v The linear function L : U U is also called linear operator, linear transformation, linear application or endomorphism. The set of all linear functions L : U V defines a linear vector space L(F). The set of all linear function call from V(F) to F (where usually F = R) L : V R, defines a dual vector space, called V (F). 3

4 Given a vector v V(F), the simplest example of dual vector v V (F) is v 1v v nv n = v,v = v,v ; f := v = [v 1 v n] Scalar product The definition of vector spaces does not include a product between elements of the space. But when a metric is necessary, usually it is the one induced by the scalar product. Given two physical vectors a = QP,b = SR, the scalar product or inner product a b is a real number defined (geometrically) as: a b = a b cosθ where a is the vector length and θ, (0 θ 180 ) is the angle between a and b; some indicate the product as a,b (see also dual spaces). Given two mathematical vectors x, y V(R) the scalar product is defined as Properties: x y = a k b k = x T y k distributive wrt sum (x+y) z = x z+y z distributive wrt product by scalar α(x y) = (αx) y = x (αy) commutative x y = y x positive x x > 0, x 0;x x = 0 iff x = 0 Geometrical (physical) definition implies the concepts of angle and length, while in the mathematical setting norm is a quantity derived by the scalar product x = x x = x 2 k = x T x k and the angle between x and y is defined as ( ) x y θ = cos 1 x y Cross product Scalar product acts between two vectors and produces a scalar; in general we would like to define a product that produces a vector. If we stay in the 3D case, we can define the cross product, aka external product, vectorial product. Given two physical 3D vectors a = QP,b = SR, the cross product c = a b is a vector orthogonal to the plane of a,b, whose length is c = a b sin θ where θ is the minimum angle that takes a to b counterclockwise (right-hand rule). Given two mathematical vectors x = [x 1 x 2 x 3 ] T, y = [y 1 y 2 y 3 ] T, with x,y R 3 the cross product x y is a vector z such that z = x y = [ ] x2 y 3 x 3 y 2 x 3 y 1 x 1 y 3 x 1 y 2 x 2 y 1 4

5 This relation can be written as [ ] x2 y 3 x 3 y 2 z = x y = x 3 y 1 x 1 y 3 x 1 y 2 x 2 y 1 = where S(x) is a antisymmetric (skew-symmetric) matrix. Cross Product Properties: Jacobi identity [ 0 x3 x 2 x 3 0 x 1 x 2 x 1 0 ][ ] y1 y 2 = S(x)y y 3 anticommutative x y = (y x) distributive wrt sum x (y+z) = (x y) distributive wrt product by a scalar α (x y) = (αx) y = x (αy) non associative x (y z) (x y) z a (b c)+b (c a)+c (a b) = 0 Given three 3D vectors x,y,z, the triple product is a non associative product: where (Grassmann identities): x (y z) (x y) z x (y z) = (x z)y (x y)z (x y) z = (x z)y (y z)x The following relation (between scalars) holds as well (x y) z = (z y) x The cross product is only defined in R 3 ; in order to generalize it in higher dimensional spaces n > 3 it is necessary to introduce the Clifford Algebras. Dyadic Product Given two vectors x, y R n, the dyadic product is defined as x 1 y 1 x 1 y n x y = xy T = D(x,y) =. x i y i. x n y 1 x n y n Properties (αx) y = x (αy) = α(x y) x (y+ z) = x y+ x z (x+y) z = x y+ x z (x y)z = x(y z) x(y z) = (x y)z Some texts call this product external product, adding noise to the nomenclature, since the external product is another type of product introduce by Grassmann. The product is non commutative: xy T yx T, since D(x,y) = D T (y,x) The matrix D has always rank ρ(d) = 1, whatever its dimension n. Relations between dyadic and cross product x (y z) = [(x z)i z x]y (x y) z = [(x z)i x z]y It is interesting to note that, while the cross product is only defined for 3D vectors, the right terms are defined for any dimension n. 5

6 Other vector products Since the scalar product does not produce a vector, and the cross product is non-commutative, but also non-associative, there was the necessity to define a product ab between vectors possessing the majority of properties of an ordinary product, i.e., the associative and distributive properties, while the commutativity was not deemed essential. Moreover the norm equality must be true, i.e., ab = a b. Many products were defined with these properties, but the two most common (in computer vision, kinematics, and quantum physics) are the Hamilton product and the Clifford product. Hamilton product Hamilton product finds its reason in the context of quaternion product. The product c = ab is axiomatically defined as c = ab = a b+ a b This product has now only an historical significance, since it has the unpleasant characteristic to produce a negative number for the product of a vector by itself aa = a a+a a = a 2 Clifford Product As reported in [04], a vector product satisfying the same axioms of the real numbers product (distributivity, associativity and commutativity) does not exist for vector spaces with dimension n 3. Dropping the commutativity axiom, it is possible to define the Clifford product (from William Clifford, ). It allows to extend the cross (external) product to vector spaces R n,n > 3. Indeed the Clifford product was already introduced some years before by Hermann Grassmann, the inventor of the exterior algebra. Bivectors Starting from R 2, given two vectors a = a 1 i+ a 2 j, and b = b 1 i+b 2 j, Clifford product is defined as: ab = a 1 b 1 + a 2 b 2 +(a 1 b 2 a 2 b 1 )e 12 = a b+(a 1 b 2 a 2 b 1 )e 12 where e 12 is a bivector and is to be understood as the signed unit area of the parallelogram with sides i and j. This is analog to the cross product i j, apart that this last is an axial vector orthogonal to the i, j plane, while the former is a so-called patch of the same plane as shown in Figure. The Clifford product is often written using the symbols introduced by Grassmann, i.e., ab = a b+ a b 6

7 Figure 5: The bivector e 12 in R 3. where a b = (a 1 b 2 a 2 b 1 )i j is the so called wedge product or exterior product that shall not be confused with the cross product a b. As said before, a b is a directed area, while a b is an axial vector. Moreover, while a b is undefined outside R 3, the wedge product can be defined for any n-dimensional space R n, where it can be interpreted as an area patch, a volume patch, a higher dimensional patch, etc. In R 3 if one assumes that the following identity holds: cc = c 2 = c c where is the scalar product assuming c = a+b, then: (a+b)(a+b) = (a+b) (a+b) aa+ab+ba+bb = a a+ a b+b a+b b a 2 + ab+ba+b 2 = a 2 + 2a b+b 2 hence and finally ab+ba = 2a b ab = 2a b ba Applications Differential geometry One of the principal applications of the exterior algebra is in differential geometry where it is used to define the bundle of differential forms on a smooth manifold. In the case of a (pseudo-)riemannian manifold, the tangent spaces come equipped with a natural quadratic form induced by the metric. Thus, one can define a Clifford bundle in analogy with the exterior bundle. This has a number of important applications in Riemannian geometry. Perhaps more importantly is the link to a spin manifold, its associated spinor bundle and spinor manifolds. Applications Physics Clifford algebras have numerous important applications in physics. Physicists usually consider a Clifford algebra to be an algebra spanned by the so-called Dirac matrices where γ 1,...,γ 4 γ i γ j + γ j γ i = 2η i j I where η = [η i j ] is the matrix of a quadratic form, having signature (p,q), typically (1,3) when working in Minkowski space (i.e., (+,,, )). The Dirac matrices were first written down by Paul Dirac when he was trying to write a relativistic first-order wave equation for the electron, and provide an explicit isomorphism from the Clifford algebra to 7

8 the algebra of complex matrices. The result was used to define the Dirac equation and introduce the Dirac operator. The entire Clifford algebra shows up in quantum field theory in the form of Dirac field bilinears. In mathematical physics, the gamma matrices, γ 1,γ 2,γ 3,γ 4, also known as the Dirac matrices, are a set of conventional matrices with specific anticommutative relations that ensure they generate a matrix representation of the Clifford algebra Cl 1,3. When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of space time acts. Spinors facilitate space-time computations in general, and in particular are fundamental to the Dirac equation for relativistic spin-1/2 particles. In Dirac representation, the four contravariant gamma matrices are γ 1 = ; γ 2 = j j γ 3 = 0 j 0 0 ; γ 4 = j Applications Computer Vision Recently, Clifford algebras have been applied in the problem of action recognition and classification in computer vision. Some authors propose a Clifford embedding to generalize traditional MACH filters to video (3D spatiotemporal volume), and vector-valued data such as optical flow. Vector-valued data is analyzed using the Clifford Fourier transform. Based on these vectors action filters are synthesized in the Clifford Fourier domain and recognition of actions is performed using Clifford Correlation. The authors demonstrate the effectiveness of the Clifford embedding by recognizing actions typically performed in classic feature films and sports broadcast television. Antisymmetric matrices Antisymmetric matrix A square matrix S is called antisymmetric or skew-symmetric when S+S T = O or S = S T A antisymmetric matrix has the following structure Therefore there it has at most n(n 1) 2 0 s 12 s 1n s 12 0 s 2n A n n = s 1n s 2n 0 independent elements. Antisymmetric matrices n(n 1) For n = 3 it results = 3, hence an antisymmetric matrix has as many element as a 3D vector v. 2 Given a vector v = [v 1 v 2 v 3 ] T it is possible to build S, and given a matrix S it is possible to extract the associated vector v. 8

9 We indicate this fact using the symbol S(v), where, by convention S(v) = [ 0 v3 ] v 2 v 3 0 v 1 v 2 v 1 0 Antisymmetric matrices Some properties: Given any vector v R 3 : S T (v) = S(v) = S( v) Given two scalars λ 1,λ 2 R: S(λ 1 u+λ 2 v) = λ 1 S(u)+λ 2 S(v) Given any two vectors v,u R 3 : S(u)v = u v = v u = S( v)u = S T (v)u Therefore S(u) is the representation of the operator (u ) and viceversa. Antisymmetric matrices The matrix S(u)S(u) = S 2 (u) is symmetrical and S 2 (u) = uu T u 2 I Hence the dyadic product can be written as D(u,u) = uu T = S 2 (u)+ u 2 I Eigenvalues and eigenvectors of antisymmetric matrices Given an antisymmetric matrix S(v), its eigenvalues are λ 1 = 0, λ 2,3 = ±j v The eigenvalue related to the eigenvector λ 1 = 0 is v; the other two are complex conjugate. The set of antisymmetric matrices is a vector space, denoted as so(3). Given two antisymmetric matrices S 1 and S 2, we call commutator or Lie bracket the following operator that is itself antisymmetric. [S 1,S 2 ] def = S 1 S 2 S 2 S 1 Antisymmetric matrices form a Lie algebra, which is related to the Lie group of orthogonal matrices. Orthogonal matrices A square matrix A R n is called orthogonal when α α 2 0 A T A = α n 9

10 with α i 0. A square orthogonal matrix U R n is called orthonormal when all the constants α i are 1: U T U = UU T = I Therefore U 1 = U T Orthonormal matrices Other properties: The columns, as well as the rows, of U are orthogonal to each other and have unit norm. U = 1; The determinant of U has unit module: det(u) = 1 therefore it can be +1 or 1. Given a vector x, its orthonormal transformation is y = Ux. Orthonormal matrices If U is an orthonormal matrix, then AU = UA = A. This property in general valid also for unitary matrices, i.e., those defined as U U = I. When U R 3 3, only 3 out of its 9 elements are independent. Scalar product is invariant to orthonormal transformations, (Ux) (Uy) = (Ux) T (Uy) = x T U T Uy = x T y = x y This means that vector lengths are invariant wrt orthonormal trasformations Ux = (Ux) T (Ux) = x T U T Ux = x T Ix = x T x = x Orthonormal matrices When considering orthonormal transformations, it is important to distinguish the two cases: When det(u) = +1, U represents a proper rotation or simply a rotation, when det(u) = 1, U represents an improper rotation or reflection. The set of rotations forms a continuous non-commutative (wrt product) group; the set of reflections do not have this quality. Intuitively this means that infinitesimal rotations exist, while infinitesimal reflections do not have any meaning. Nonetheless, reflections are the most basic transformation in 3D spaces, in the sense that translations, rotations and roto-reflections (slidings) are obtained from the composition of two or three reflections 10

11 Figure 6: Reflections producing rotations and translation in R 2. Orthonormal matrices If U is an orthonormal matrix, the distributive property wrt the cross product holds: (with general A matrices this is not true). U(x y) = (Ux) (Uy) For any proper rotation matrix U and a generic vector x the following holds US(x)U T y = U ( x (U T y) ) = (Ux) (UU T y) = (Ux) y = S(Ux)y where S(x) is the antisymmetric matrix associated with x; therefore: US(x)U T = S(Ux) US(x) = S(Ux)U References [01] S.L. Altmann, Icons and Symmetries, Clarendon Press, [02] Van Der Ha, Shuster, A Tutorial on Vectors and Attitude. [03] K. Jänich, Linear Algebra, Springer, [04] J. Stillwell, Mathematics and Its History, Springer, [05] J. Gallier, Geometric Methods and Applications, Springer,