The roots of a split quaternion

Özdemir, Mustafa

doi:10.1016/j.aml.2008.03.020

Cited by 78 publications

(51 citation statements)

References 7 publications

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“…The existence of matrix square root here is related to matrix positive eigenvalues. In the context of noncommutative Clifford algebra (CA), the main attempts up till now were concentrated on the square roots of quaternions [19,20] or their derivatives such as coquaternions (also called split quaternions) or nectarines [9,20,21]. The square root of biquaternion (complex quaternion) was considered in [22].…”

Section: Introductionmentioning

confidence: 99%

Square root of a multivector in 3D Clifford algebras

Dargys

Acus

2020

NAMC

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The problem of square root of multivector (MV) in real 3D (n = 3) Clifford algebras Cl 3,0, Cl 2,1, Cl 1,2 and Cl 0,3 is considered. It is shown that the square root of general 3D MV can be extracted in radicals. Also, the article presents basis-free roots of MV grades such as scalars, vectors, bivectors, pseudoscalars and their combinations, which may be useful in applied Clifford algebras. It is shown that in mentioned Clifford algebras, there appear isolated square roots and continuum of roots on hypersurfaces (infinitely many roots). Possible numerical methods to extract square root from the MV are discussed too. As an illustration, the Riccati equation formulated in terms of Clifford algebra is solved.The Clifford algebra Cl p,q is an associative noncommutative algebra, where (p, q) indicates vector space metric. In 3D case, the MV consists of the following elements (basis blades) {1, e 1 , e 2 , e 3 , e 12 , e 13 , e 23 , e 123 ≡ I}, where e i are the orthogonal basis vectors, and e ij are the bivectors (oriented planes). The last term is called the pseudoscalar. The shorthand notation, for example, e ij means e ij = e i • e j , where the circle indicates the geometric or Clifford product. Usually, the multiplication symbol is omitted, and one writes e ij = e i e j . The number of subscripts indicates the grade of basis element, so that the scalar is a grade-0 element, the vectors are the grade-1 elements etc. In the orthonormalized basis, the products of vectors satisfies e i e j + e j e i = ±2δ ij .

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Section: Introductionmentioning

confidence: 99%

Square root of a multivector in 3D Clifford algebras

Dargys

Acus

2020

NAMC

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“…On the other hand, quaternions over finite rings have attracted significant attention since they have applications in coding theory see [7,8,11].…”

Section: Introductionmentioning

confidence: 99%

On the Structure of Quaternion Rings Over $${\mathbb{Z}/n\mathbb{Z}}$$ Z / n Z

Grau

Miguel

Oller-Marcén

2015

Adv. Appl. Clifford Algebras

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“…The algebra H(1, −1) is called the algebra of coquaternions [2] or also called the algebra of para-quaternions [8], or the algebra of split-quaternions [5], [16], or the algebra of anti-quaternions, or the algebra of pseudo-quaternions [18, p. 389], or hyperbolic quaternions [1]. We denote by {1, i 1 , i 2 , i 3 } the basis of coquaternion algebra.…”

Section: Introductionmentioning

confidence: 99%

“…The norm of an octonion a ∈ O(1, 1, −1), a = a 0 + a 1 f 1 + a 2 f 2 + a 3 f 3 + a 4 f 4 + a 5 f 5 + a 6 f 6 + a 7 f 7 , in this situation, can be positive, zero or negative. In the following, we used definitions and propositions obtained for the split quaternions as in [16] to generalized them to similar results for the split octonions. A split octonion is called spacelike, timelike or lightlike if N (a) < 0, N (a) > 0 or N (a) = 0.…”

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An Efficient Method for Solving Equations in Generalized Quaternion and Octonion Algebras

Flaut

Shpakivskyi

2014

Adv. Appl. Clifford Algebras

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Quaternions often appear in wide areas of applied science and engineering such as wireless communications systems, mechanics, etc. It is known that are two types of non-isomorphic generalized quaternion algebras, namely: the algebra of quaternions and the algebra of coquaternions. In this paper, we present the formulae to pass from a basis in the generalized quaternion algebras to a basis in the division quaternions algebra or to a basis in the coquaternions algebra and vice versa. The same result was obtained for the generalized octonion algebra. Moreover, we emphasize the applications of these results to the algebraic equations and De Moivre's formula in generalized quaternion algebras and in generalized octonion division algebras.

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The roots of a split quaternion

Cited by 78 publications

References 7 publications

Square root of a multivector in 3D Clifford algebras

Square root of a multivector in 3D Clifford algebras

On the Structure of Quaternion Rings Over $${\mathbb{Z}/n\mathbb{Z}}$$ Z / n Z

An Efficient Method for Solving Equations in Generalized Quaternion and Octonion Algebras

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