Multivectors

Multivectors

A multivector is a mathematical object that belongs to the Geometric Algebra. It is a linear combination of basis blades defined in the algebra and can be expressed as a sum of scalars, vectors, bivectors, trivectors, and so on. In general, a multivector can have multiple components and can represent various geometric entities such as points, lines, planes, and volumes.From: Introduction to Geometric Algebra Computing [2020], Efficient ray-tracing procedure for radio wave propagation modeling using homogeneous geometric algebra [2020]

Mathematical Basics and 2D Euclidean Geometric Algebra

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Published in Dietmar Hildenbrand, Introduction to Geometric Algebra Computing, 2020

Dietmar Hildenbrand

While the pairwise orthogonal and normalized basis vectors e1, e2,…, en are the basic algebraic elements of an n-dimensional vector algebra, they are only one part of the algebraic elements of an n-dimensional Geometric Algebra1. Blades are the basic algebraic elements of Geometric Algebra. An n-dimensional Geometric Algebra consists of blades with grades 0, 1, 2, …, n, where a scalar is a 0-blade (a blade of grade 0) and the 1-blades are the basis vectors e1, e2,…, en. The 2-blades2ei ∧ ej are blades spanned by two 1-blades, and so on. There exists only one element of the maximum grade n, I = e1 ∧ e2…∧en. It is therefore also called the pseudoscalar. A linear combination of k-blades is called a k-vector (or a vector, bivector, trivector….). A linear combination of blades with different grades is called a multivector. Multivectors are the general elements of a Geometric Algebra.

Efficient ray-tracing procedure for radio wave propagation modeling using homogeneous geometric algebra

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Published in Electromagnetics, 2020

Ahmad H. Eid, Heba Y. Soliman, Sherif M. Abuelenin

A general multivector is simply a linear combination of basis blades defined in . If is a linear combination of elements in , it is called a multivector of grade , or a -vector for short. A general multivector is typically composed of 4 parts each with a different grade value from 0 to 4. The grade extraction operator extracts the -vector part of having grade , where for all or by definition. For example, if then where , , , , and .

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Mathematical Basics and 2D Euclidean Geometric Algebra

Efficient ray-tracing procedure for radio wave propagation modeling using homogeneous geometric algebra