In this paper, we established the connection between generalized quaternion algebra and real (complex) matrix algebras by using Hamilton operators. We obtained complex matrices corresponding to basis of quaternions. Also, investigated features matrices. get Pauli Then, have shown that produced these is isomorphic Clifford Cl(E_αβ^3) space E_αβ^3.
 Finally, studied relations among symplecti...

1.1. The matrix Ξkk′ In the main text, we are given two unit vectors μ1k and μ2k′ in R. We define Ξkk′ = Ξ(μ1k, μ2k′), where Ξ(u, v) ∈ R4×4 is defined by u (q ◦ v) = qΞ(u, v)q, where u = (ui, uj , uk), v = (vi, vj , vk), and q = (qi, qj , qk, qr). By standard quaternion rotation formula, we have u (q ◦ v) = ui uj uk T 1− 2q j − 2q k 2(qiqj − qkqr) 2(qiqk + qjqr) 2(qiqj + qkqr) 1− 2q i − 2...

In a quaternion structure composed of four real dimensions, we derive form wherein three complex numbers are combined. Thereafter, examined whether this includes the algebraic properties and transformations were necessary for its application to system. addition, defined regular function in quaternions, expressed as combination numbers. Furthermore, derived Cauchy-Riemann equation investigate qu...

Rigid body dynamics on the rotation group have typically been represented in terms of rotation matrices, unit quaternions, or local coordinates, such as Euler angles. Due to the coordinate singularities associated with local coordinate charts, it is common in engineering applications to adopt the unit quaternion representation, and the numerical simulations typically impose the unit length cond...

Iztok Fister and Iztok Fister Jr. Abstract This paper introduces a novel idea for representation of individuals using quaternions in swarm intelligence and evolutionary algorithms. Quaternions are a number system, which extends complex numbers. They are successfully applied to problems of theoretical physics and to those areas needing fast rotation calculations. We propose the application of qu...

Quaternions are ordered quadruples of four numbers subject to specified rules of addition and multiplication, which can represent points in four-dimensional (4D) space and which form finite groups under multiplication isomorphic to polyhedral groups. Projection of the 8 quaternions of the dihedral group D2h, with only two-fold symmetry, into 3D space provides a basis for crystal lattices up to ...