Geometric matrix algebra

Sobczyk, Garret

doi:10.1016/j.laa.2007.06.015

Cited by 37 publications

(19 citation statements)

References 3 publications

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“…Using the Eigen [23] C++ matrix library the function in Listing 1 can be implemented as presented in Listing 2 and Listing 3. Further, the conformal model can be implemented using Vahlen matrices, see [13] and [43]. This approach is relatively simple to implement for the Euclidean model as all elements in the geometric algebra are represented by matrices.…”

Section: Automatic Multivector Differentiationmentioning

confidence: 99%

Automatic Multivector Differentiation and Optimization

Tingelstad

Egeland

2016

Adv. Appl. Clifford Algebras

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Section: Automatic Multivector Differentiationmentioning

confidence: 99%

Automatic Multivector Differentiation and Optimization

Tingelstad

Egeland

2016

Adv. Appl. Clifford Algebras

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“…The complex vectors Z = a + i b and its conjugateZ = a − i b defined in the Equations (13) and (14) actually form a physical space when the third direction is chosen normal to the plane containing the orthogonal vetors a and b. Let c = 2i (a ∧ b) be a vector normal to both a and b.…”

Section: Complex Vector Spacementioning

confidence: 99%

“…The geometric center is defined as that it consists of those elements of the algebra which commute with every element of the algebra [14]. As the pseudoscalar i commutes with all the elements of the algebra, it is called the geometric center of the algebra.…”

Section: Geometric Algebra Of Euclidean Spacementioning

confidence: 99%

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Algebra of Complex Vectors and Applications in Electromagnetic Theory and Quantum Mechanics

Muralidhar

2015

Mathematics

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A complex vector is a sum of a vector and a bivector and forms a natural extension of a vector. The complex vectors have certain special geometric properties and considered as algebraic entities. These represent rotations along with specified orientation and direction in space. It has been shown that the association of complex vector with its conjugate generates complex vector space and the corresponding basis elements defined from the complex vector and its conjugate form a closed complex four dimensional linear space. The complexification process in complex vector space allows the generation of higher n-dimensional geometric algebra from (n − 1)-dimensional algebra by considering the unit pseudoscalar identification with square root of minus one. The spacetime algebra can be generated from the geometric algebra by considering a vector equal to square root of plus one. The applications of complex vector algebra are discussed mainly in the electromagnetic theory and in the dynamics of an elementary particle with extended structure. Complex vector formalism simplifies the expressions and elucidates geometrical understanding of the basic concepts. The analysis shows that the existence of spin transforms a classical oscillator into a quantum oscillator. In conclusion the classical mechanics combined with zeropoint field leads to quantum mechanics.

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“…Similarly, different kinds of 2-dimensional numbers can be constructed by modifying the multiplication rule. For example, split-complex numbers correspond to i 2 = 1 (also called hyperbolic numbers) and dual complex numbers correspond to i 2 = 0 [13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Coset Group Construction of Multidimensional Number Systems

Petrache

2014

Symmetry

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Extensions of real numbers in more than two dimensions, in particular quaternions and octonions, are finding applications in physics due to the fact that they naturally capture symmetries of physical systems. However, in the conventional mathematical construction of complex and multicomplex numbers multiplication rules are postulated instead of being derived from a general principle. A more transparent and systematic approach is proposed here based on the concept of coset product from group theory. It is shown that extensions of real numbers in two or more dimensions follow naturally from the closure property of finite coset groups adding insight into the utility of multidimensional number systems in describing symmetries in nature.

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Geometric matrix algebra

Cited by 37 publications

References 3 publications

Automatic Multivector Differentiation and Optimization

Automatic Multivector Differentiation and Optimization

Algebra of Complex Vectors and Applications in Electromagnetic Theory and Quantum Mechanics

Coset Group Construction of Multidimensional Number Systems

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