# Multivector

In multilinear algebra, a multivector or Clifford number[1] is an element of the (graded) exterior algebra on a vector space, Λ*V. This algebra consists of linear combinations of simple k-vectors (also known as decomposable k-vectors or k-blades)

$v_1\wedge\cdots\wedge v_k.$

"Multivector" may mean either homogeneous elements (all terms of the sum have the same grade or degree k), which are referred to as k-vectors or p-vectors,[2] or may allow sums of terms in different degrees.

The k-th exterior power,

$\Lambda^k(V),$

is the vector space of formal sums of k-multivectors. The product of a k-multivector and an -multivector is a (k + )-multivector. So, the direct sum $\bigoplus_k \Lambda^k(V)$ forms an associative algebra, which is closed with respect to the wedge product. This algebra, commonly denoted by Λ(V), is called the exterior algebra of V.

In differential geometry, a p-vector is the tensor obtained by taking linear combinations of the wedge product of p tangent vectors, for some integer p ≥ 0. It is the dual concept to a p-form.

For p = 0, 1, 2 and 3, these are often called respectively scalars, vectors, bivectors and trivectors; they are respectively dual to 0-forms, 1-forms, 2-forms and 3-forms.[3][4]

## Examples

Geometric interpretation for the exterior product of n vectors (u, v, w) to obtain an n-vector (parallelotope elements), where n = grade,[5] for n = 1, 2, 3. The "circulations" show orientation.[6]

In the presence of a volume form (such as given an inner product and an orientation), pseudovectors and pseudoscalars can be identified with vectors and scalars, which is routine in vector calculus, but without a volume form this cannot be done without a choice.

In the Algebra of physical space (the geometric algebra of Euclidean 3-space, used as a model of 3+1 spacetime), a sum of a scalar and a vector is called a paravector, and represents a point in spacetime (the vector the space, the scalar the time).

### Bivectors

A bivector is therefore an element of the antisymmetric tensor product of a tangent space with itself.

In geometric algebra, also, a bivector is a grade 2 element (a 2-vector) resulting from the wedge product of two vectors, and so it is geometrically an oriented area, in the same way a vector is an oriented line segment. If a and b are two vectors, the bivector a ∧ b has

• a norm which is its area, given by
$\Vert \mathbf a \wedge \mathbf b \Vert = \Vert \mathbf{a} \Vert \, \Vert \mathbf{b} \Vert \, \sin(\phi_{a,b})$
• a direction: the plane where that area lies on, i.e., the plane determined by a and b, as long as they are linearly independent;
• an orientation (out of two), determined by the order in which the originating vectors are multiplied.

Bivectors are connected to pseudovectors, and are used to represent rotations in geometric algebra.

As bivectors are elements of a vector space Λ2V (where V is a finite-dimensional vector space with $\dim V =n$), it makes sense to define an inner product on this vector space as follows. First, write any element F ∈ Λ2V in terms of a basis (eiej)1 ≤ i < jn of Λ2V as

$F = F^{ab} e_a \wedge e_b \quad (1 \le a < b \le n)$

where the Einstein summation convention is being used.

Now define a map G : Λ2V × Λ2VR by insisting that

$G(F, H) := \, G_{abcd}F^{ab}H^{cd}$

where $G_{abcd}$ are a set of numbers.

## Geometric algebra

In geometric algebra, a multivector is defined to be the sum of different-grade k-blades, such as the summation of a scalar, a vector, and a 2-vector.[7] A sum of only k-grade components is called a k-vector,[8] or a homogeneous multivector.[9]

The highest grade element in a space is called a pseudoscalar.

If a given element is homogeneous of a grade k, then it is a k-vector, but not necessarily a k-blade. Such an element is a k-blade when it can be expressed as the wedge product of k vectors. A geometric algebra generated by a 4-dimensional Euclidean vector space illustrates the point with an example: The sum of any two blades with one taken from the XY-plane and the other taken from the ZW-plane will form a 2-vector that is not a 2-blade. In a geometric algebra generated by a Euclidean vector space of dimension 2 or 3, all sums of 2-blades may be written as a single 2-blade.

## Applications

Bivectors play many important roles in physics, for example, in the classification of electromagnetic fields.

## Notes

1. ^ John Snygg (2012), A New Approach to Differential Geometry Using Clifford’s Geometric Algebra, Birkhäuser, p.5 §2.12
2. ^ Élie Cartan, The theory of spinors, p. 16, considers only homogeneous vectors, particularly simple ones, referring to them as "multivectors" (collectively) or p-vectors (specifically).
3. ^ William M Pezzaglia Jr. (1992). "Clifford algebra derivation of the characteristic hypersurfaces of Maxwell's equations". In Julian Ławrynowicz. Deformations of mathematical structures II. Springer. p. 131 ff. ISBN 0-7923-2576-1. "Hence in 3D we associate the alternate terms of pseudovector for bivector, and pseudoscalar for the trivector"
4. ^ Baylis (1994). Theoretical methods in the physical sciences: an introduction to problem solving using Maple V. Birkhäuser. p. 234, see footnote. ISBN 0-8176-3715-X.
5. ^ R. Penrose (2007). The Road to Reality. Vintage books. ISBN 0-679-77631-1.
6. ^ J.A. Wheeler, C. Misner, K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. p. 83. ISBN 0-7167-0344-0.
7. ^ Marcos A. Rodrigues (2000). "§1.2 Geometric algebra: an outline". Invariants for pattern recognition and classification. World Scientific. p. 3 ff. ISBN 981-02-4278-6.
8. ^ R Wareham, J Cameron & J Lasenby (2005). "Applications of conformal geometric algebra in computer vision and graphics". In Hongbo Li, Peter J. Olver, Gerald Sommer. Computer algebra and geometric algebra with applications. Springer. p. 330. ISBN 3-540-26296-2.
9. ^ Eduardo Bayro-Corrochano (2004). "Clifford geometric algebra: A promising framework for computer vision, robotics and learning". In Alberto Sanfeliu, José Francisco Martínez Trinidad, Jesús Ariel Carrasco Ochoa. Progress in pattern recognition, image analysis and applications. Springer. p. 25. ISBN 3-540-23527-2.