The theory of quaternions was introduced in the mid nineteenth century, and it found many applications in classical mechanics, quantum mechanics, and the theory of relativity. Quaternions were also later used in aerospace applications and flight simulators, particularly when inertial attitude referencing and related control schemes where employed. However, it is only in the recent past that gra...

Some complex quaternionic equations in the type AX - XB = C are investigated. For convenience, these equations were called generalized Sylvester-quaternion equations, which include the Sylvester equation as special cases. By the real matrix representations of complex quaternions, the necessary and sufficient conditions for the solvability and the general expressions of the solutions are obtained.

The well-known canonical coherent states are expressed as an infinite series in powers of a complex number z and a positive sequence of real numbers ρ(m) = m!. In this article, in analogy with the canonical coherent states, we present a class of vector coherent states by replacing the complex variable z by a real Clifford matrix. We also present another class of vector coherent states by simult...

Since quaternions have isomorphic representations in matrix form we investigate various well known matrix decompositions for quaternions.

8 Vectors and Quaternions 145 8.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 8.2 Displacement Vectors . . . . . . . . . . . . . . . . . . . . . . . 145 8.3 The Parallelogram Law of Vector Addition . . . . . . . . . . . 146 8.4 The Length of Vectors . . . . . . . . . . . . . . . . . . . . . . 147 8.5 Scalar Multiples of Vectors . . . . . . . . . . . . . . . . . . ....

The widespread use of unmanned aerial vehicles during warfare has intensified the problem their management, especially when they are used in large groups. One main tasks is to ensure coordinated movement group's aircraft space. Optimizing each device group three-dimensional space expedient mathematical models. any vehicle can be presented as a combination translational and rotational movements,...

8 Vectors and Quaternions 40 8.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 8.2 Displacement Vectors . . . . . . . . . . . . . . . . . . . . . . . 40 8.3 The Parallelogram Law of Vector Addition . . . . . . . . . . . 41 8.4 The Length of Vectors . . . . . . . . . . . . . . . . . . . . . . 42 8.5 Scalar Multiples of Vectors . . . . . . . . . . . . . . . . . . . . 43...

Abstract Quaternions are a four-dimensional hypercomplex number system discovered by Hamilton in 1843 and next intensively applied mathematics, modern physics, computer graphics other fields. After the discovery of quaternions, modified quaternions were also defined such way that commutative property multiplication is possible. That called as studied used for example signal processing. In this ...

We reformulate Special Relativity by a quaternionic algebra on reals. Using real linear quaternions, we show that previous difficulties, concerning the appropriate transformations on the 3 + 1 space-time, may be overcome. This implies that a complexified quaternionic version of Special Relativity is a choice and not a necessity. a) e-mail: [email protected]

We compute the intrinsic volumes of the cone of positive semidefinite matrices over the real numbers, over the complex numbers, and over the quaternions, in terms of integrals related to Mehta’s integral. Several applications for the probabilistic analysis of semidefinite programming are given. AMS subject classifications: 15B48, 52A55, 53C65, 60D05, 90C22